 8.8.4.1: (a) What is an alternating series? 18. p (b) Under what conditions ...
 8.8.5.1: What is a power series?
 8.8.6.1: If the radius of convergence of the power series is 10, what is the...
 8.8.7.1: If for all , write a formula for .
 8.8.1.1: (a) What is a sequence? (b) What does it mean to say that ? (c) Wha...
 8.8.8.1: (a) Find the Taylor polynomials up to degree 6 for centered at . Gr...
 8.8.2.1: (a) What is the difference between a sequence and a series? x (b) W...
 8.1: (a) What is a convergent sequence? (b) What is a convergent series?...
 8.8.3.1: Draw a picture to show that What can you conclude about the series?
 8.8.4.2: What can you say about the series in each of the following cases? (...
 8.8.5.2: (a) What is the radius of convergence of a power series? How do you...
 8.8.6.2: If the radius of convergence of the power series is 10, what is the...
 8.8.7.2: The graph of is shown. (a) Explain why the series is not the Taylor...
 8.8.1.2: (a) What is a convergent sequence? Give two examples. (b) What is a...
 8.8.8.2: (a) Find the Taylor polynomials up to degree 3 for centered at . Gr...
 8.8.2.2: Explain what it means to say that .
 8.2: (a) What is a bounded sequence? (b) What is a monotonic sequence? (...
 8.8.3.2: Suppose is a continuous positive decreasing function for and . By d...
 8.8.6.3: Find a power series representation for the function and determine t...
 8.8.7.3: If for find the Maclaurin series for and its radius of convergence.
 8.8.1.3: List the first six terms of the sequence defined by Does the sequen...
 8.8.8.3: Find the Taylor polynomial for the function at the number . Graph a...
 8.3: (a) What is a geometric series? Under what circumstances is it conv...
 8.8.3.3: Suppose and are series with positive terms and is known to be conve...
 8.8.6.4: Find a power series representation for the function and determine t...
 8.8.7.4: Find the Taylor series for centered at 4 if What is the radius of c...
 8.8.1.4: List the first nine terms of the sequence . Does 31. an this sequen...
 8.8.8.4: Find the Taylor polynomial for the function at the number . Graph a...
 8.4: Suppose and is the partial sum of the series. What is ? What is ?
 8.8.3.4: Suppose and are series with positive terms and is known to be diver...
 8.8.6.5: Find a power series representation for the function and determine t...
 8.8.7.5: Find the Maclaurin series for using the definition of a Maclaurin s...
 8.8.2.5: Determine whether the geometric series is convergent or divergent. ...
 8.8.8.5: Find the Taylor polynomial for the function at the number . Graph a...
 8.5: State the following. (a) The Test for Divergence (b) The Integral T...
 8.8.3.5: It is important to distinguish between and What name is given to th...
 8.8.4.6: Test the series for convergence or divergence.n1 1 n sn 1 2sn
 8.8.6.6: Find a power series representation for the function and determine t...
 8.8.7.6: Find the Maclaurin series for using the definition of a Maclaurin s...
 8.8.2.6: Determine whether the geometric series is convergent or divergent. ...
 8.8.8.6: Find the Taylor polynomial for the function at the number . Graph a...
 8.6: (a) What is an absolutely convergent series? (b) What can you say a...
 8.8.3.6: Use the Integral Test to determine whether the series is convergent...
 8.8.4.7: Test the series for convergence or divergence.n1 1 n 3n 1 2n 1 7.
 8.8.6.7: Find a power series representation for the function and determine t...
 8.8.7.7: Find the Maclaurin series for using the definition of a Maclaurin s...
 8.8.1.7: Find a formula for the general term of the sequence, assuming that ...
 8.8.2.7: Determine whether the geometric series is convergent or divergent. ...
 8.8.8.7: Find the Taylor polynomial for the function at the number . Graph a...
 8.7: If a series is convergent by the Alternating Series Test, how do yo...
 8.8.3.7: Use the Integral Test to determine whether the series is convergent...
 8.8.4.8: Test the series for convergence or divergence.n1 1 n1 ln n n
 8.8.5.8: Find the radius of convergence and interval of convergence of the s...
 8.8.6.8: Find a power series representation for the function and determine t...
 8.8.7.8: Find the Maclaurin series for using the definition of a Maclaurin s...
 8.8.2.8: Determine whether the geometric series is convergent or divergent. ...
 8.8.8.8: Find the Taylor polynomial for the function at the number . Graph a...
 8.8: (a) Write the general form of a power series. (b) What is the radiu...
 8.8.3.8: Use the Integral Test to determine whether the series is convergent...
 8.8.4.9: Show that the series is convergent. How many terms of the series do...
 8.8.5.9: Find the radius of convergence and interval of convergence of the s...
 8.8.6.9: Find a power series representation for the function and determine t...
 8.8.7.9: Find the Maclaurin series for using the definition of a Maclaurin s...
 8.8.1.9: Determine whether the sequence converges or diverges. If it converg...
 8.8.2.9: Determine whether the series is convergent or divergent. If it is c...
 8.9: Suppose is the sum of a power series with radius of convergence . (...
 8.8.3.9: Use the Comparison Test to determine whether the series is converge...
 8.8.4.10: Show that the series is convergent. How many terms of the series do...
 8.8.5.10: Find the radius of convergence and interval of convergence of the s...
 8.8.6.10: Find a power series representation for the function and determine t...
 8.8.7.10: Find the Maclaurin series for using the definition of a Maclaurin s...
 8.8.1.10: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.10: (a) Approximate by a Taylor polynomial with degree at the number . ...
 8.8.2.10: Determine whether the series is convergent or divergent. If it is c...
 8.10: (a) Write an expression for the thdegree Taylor polynomial of cent...
 8.8.3.10: Use the Comparison Test to determine whether the series is converge...
 8.8.4.11: Show that the series is convergent. How many terms of the series do...
 8.8.5.11: Find the radius of convergence and interval of convergence of the s...
 8.8.7.11: Find the Taylor series for centered at the given value of . [Assume...
 8.8.1.11: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.11: (a) Approximate by a Taylor polynomial with degree at the number . ...
 8.8.2.11: Determine whether the series is convergent or divergent. If it is c...
 8.11: Write the Maclaurin series and the interval of convergence for each...
 8.8.3.11: Determine whether the series is convergent or divergent.1 1 8 1 27 ...
 8.8.4.12: Show that the series is convergent. How many terms of the series do...
 8.8.5.12: Find the radius of convergence and interval of convergence of the s...
 8.8.7.12: Find the Taylor series for centered at the given value of . [Assume...
 8.8.1.12: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.12: (a) Approximate by a Taylor polynomial with degree at the number . ...
 8.8.2.12: Determine whether the series is convergent or divergent. If it is c...
 8.12: Write the binomial series expansion of . What is the radius of conv...
 8.8.3.12: Determine whether the series is convergent or divergent.n1 5 n4 4 n...
 8.8.4.13: Approximate the sum of the series correct to four decimal places.n ...
 8.8.5.13: Find the radius of convergence and interval of convergence of the s...
 8.8.6.13: (a) Use differentiation to find a power series representation for W...
 8.8.7.13: Find the Taylor series for centered at the given value of . [Assume...
 8.8.1.13: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.13: (a) Approximate by a Taylor polynomial with degree at the number . ...
 8.8.2.13: Determine whether the series is convergent or divergent. If it is c...
 8.13: If converges for all , then .
 8.8.3.13: Determine whether the series is convergent or divergent.n1 nen 13.
 8.8.4.14: Approximate the sum of the series correct to four decimal places.n1...
 8.8.5.14: Find the radius of convergence and interval of convergence of the s...
 8.8.6.14: (a) Find a power series representation for . What is the radius of ...
 8.8.7.14: Find the Taylor series for centered at the given value of . [Assume...
 8.8.1.14: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.14: (a) Approximate by a Taylor polynomial with degree at the number . ...
 8.8.2.14: Determine whether the series is convergent or divergent. If it is c...
 8.14: If and are divergent, then is divergent.
 8.8.3.14: Determine whether the series is convergent or divergent.n1 n2 n3 1
 8.8.4.15: Approximate the sum of the series correct to four decimal places.n1...
 8.8.5.15: Find the radius of convergence and interval of convergence of the s...
 8.8.6.15: Find a power series representation for the function and determine t...
 8.8.7.15: Find the Taylor series for centered at the given value of . [Assume...
 8.8.1.15: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.15: (a) Approximate by a Taylor polynomial with degree at the number . ...
 8.8.2.15: Determine whether the series is convergent or divergent. If it is c...
 8.15: If and are divergent, then is divergent.
 8.8.3.15: Determine whether the series is convergent or divergent.16. n2 1 n ...
 8.8.4.16: Approximate the sum of the series correct to four decimal places.n1...
 8.8.5.16: Find the radius of convergence and interval of convergence of the s...
 8.8.6.16: Find a power series representation for the function and determine t...
 8.8.7.16: Find the Taylor series for centered at the given value of . [Assume...
 8.8.1.16: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.16: (a) Approximate by a Taylor polynomial with degree at the number . ...
 8.8.2.16: Determine whether the series is convergent or divergent. If it is c...
 8.16: If is decreasing and for all , then is convergent.
 8.8.3.16: Determine whether the series is convergent or divergent.n1 n2 1 3n4...
 8.8.4.17: Is the 50th partial sum of the alternating series an overestimate o...
 8.8.5.17: Find the radius of convergence and interval of convergence of the s...
 8.8.6.17: Find a power series representation for the function and determine t...
 8.8.7.17: Find the Taylor series for centered at the given value of . [Assume...
 8.8.1.17: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.17: Use the information from Exercise 3 to estimate correct to five dec...
 8.8.2.17: Determine whether the series is convergent or divergent. If it is c...
 8.17: If and converges, then converges.
 8.8.3.17: Determine whether the series is convergent or divergent.n1 cos2 n n2 1
 8.8.4.18: For what values of is the following series convergent?n1 1 n1 np 1. (
 8.8.5.18: Find the radius of convergence and interval of convergence of the s...
 8.8.6.18: Find a power series representation for the function and determine t...
 8.8.7.18: Find the Taylor series for centered at the given value of . [Assume...
 8.8.1.18: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.18: Use the information from Exercise 12 to estimate correct to five de...
 8.8.2.18: Determine whether the series is convergent or divergent. If it is c...
 8.18: If and , then .
 8.8.3.18: Determine whether the series is convergent or divergent.n1 4 3n 2n
 8.8.4.19: Determine whether the series is absolutely convergent, conditionall...
 8.8.5.19: If is convergent, does it follow that the following series are conv...
 8.8.6.19: Find a power series representation for , and graph and several part...
 8.8.7.19: Prove that the series obtained in Exercise 5 represents for all .
 8.8.1.19: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.19: Use Taylors Formula to determine the number of terms of the Maclaur...
 8.8.2.19: Determine whether the series is convergent or divergent by expressi...
 8.19: Determine whether the series is convergent or divergent.n1 1 n1 sn n 1
 8.8.3.19: Determine whether the series is convergent or divergent.n1 n 1 n4n
 8.8.4.20: Determine whether the series is absolutely convergent, conditionall...
 8.8.5.20: Suppose that converges when and A diverges when . What can be said ...
 8.8.6.20: Find a power series representation for , and graph and several part...
 8.8.7.20: Prove that the series obtained in Exercise 16 represents for all .
 8.8.1.20: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.20: Suppose you know that and the Taylor series of centered at 4 conver...
 8.8.2.20: Determine whether the series is convergent or divergent by expressi...
 8.20: Determine whether the series is convergent or divergent.n1 sn 1 sn 1 n
 8.8.3.20: Determine whether the series is convergent or divergent.n1 1 sn3 1
 8.8.4.21: Determine whether the series is absolutely convergent, conditionall...
 8.8.5.21: If is a positive integer, find the radius of convergence of the ser...
 8.8.6.21: Find a power series representation for , and graph and several part...
 8.8.7.21: Prove that the series obtained in Exercise 9 represents sinh x for ...
 8.8.1.21: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.21: Use the Alternating Series Estimation Theorem or Taylors Formula to...
 8.8.2.21: Determine whether the series is convergent or divergent by expressi...
 8.21: Determine whether the series is conditionally convergent, absolutel...
 8.8.3.21: Determine whether the series is convergent or divergent.n1 1 sn2 1 2
 8.8.4.22: Determine whether the series is absolutely convergent, conditionall...
 8.8.5.22: Graph the first several partial sums of the series , together with ...
 8.8.6.22: Find a power series representation for , and graph and several part...
 8.8.7.22: Prove that the series obtained in Exercise 10 represents for all .
 8.8.1.22: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.22: Use the Alternating Series Estimation Theorem or Taylors Formula to...
 8.8.2.22: Determine whether the series is convergent or divergent by expressi...
 8.22: Determine whether the series is conditionally convergent, absolutel...
 8.8.3.22: Determine whether the series is convergent or divergent.n1 1 2n 3
 8.8.4.23: Determine whether the series is absolutely convergent, conditionall...
 8.8.5.23: The function defined by is called the Bessel function of order 1. (...
 8.8.6.23: Evaluate the indefinite integral as a power series. What is the rad...
 8.8.1.23: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.23: A car is moving with speed 20 ms and acceleration 2 ms at a given i...
 8.8.2.23: Express the number as a ratio of integers.0.2 0.2222...
 8.23: Determine whether the series is conditionally convergent, absolutel...
 8.8.3.23: Determine whether the series is convergent or divergent.n n1 2 1 n nsn
 8.8.4.24: Determine whether the series is absolutely convergent, conditionall...
 8.8.5.24: The function defined by is called the Airy function after the Engli...
 8.8.6.24: Evaluate the indefinite integral as a power series. What is the rad...
 8.8.1.24: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.24: The resistivity of a conducting wire is the reciprocal of the condu...
 8.8.2.24: Express the number as a ratio of integers.0.73 0.73737373 . . .
 8.24: Determine whether the series is conditionally convergent, absolutel...
 8.8.3.24: Determine whether the series is convergent or divergent.n0 1 sin n ...
 8.8.3.25: Determine whether the series is convergent or divergent.2 n1 sin 1 n
 8.8.4.25: Determine whether the series is absolutely convergent, conditionall...
 8.8.5.25: A function is defined by that is, its coefficients are and for all ...
 8.8.6.25: Evaluate the indefinite integral as a power series. What is the rad...
 8.8.1.25: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.25: An electric dipole consists of two electric charges of equal magnit...
 8.8.2.25: Express the number as a ratio of integers.3.417 3.417417417...
 8.25: Find the sum of the series.n1 22n1 5n
 8.8.3.26: Determine whether the series is convergent or divergent.n1 n 5 s 3 ...
 8.8.4.26: Determine whether the series is absolutely convergent, conditionall...
 8.8.5.26: If , where for all , find the interval of convergence of the series...
 8.8.6.26: Evaluate the indefinite integral as a power series. What is the rad...
 8.8.1.26: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.26: If a water wave with length moves with velocity across a body of wa...
 8.8.2.26: Express the number as a ratio of integers.6.254 6.2545454 . . .
 8.26: Find the sum of the series. n1 1 nn 3
 8.8.3.27: Find the values of for which the series is convergent. p n2 1 nln n...
 8.8.4.27: Determine whether the series is absolutely convergent, conditionall...
 8.8.5.27: Show that if , where , then the radius of convergence of the power ...
 8.8.6.27: Use a power series to approximate the definite integral to six deci...
 8.8.1.27: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.27: If a surveyor measures differences in elevation when making plans f...
 8.8.2.27: Find the values of for which the series converges. Find the sum of ...
 8.27: Find the sum of the series.n1 tan1 n 1 tan1 n 18 D
 8.8.3.28: Find the values of for which the series is convergent. n1 ln n n p
 8.8.4.28: Determine whether the series is absolutely convergent, conditionall...
 8.8.5.28: Suppose that the power series satisfies for all . Show that if exis...
 8.8.6.28: Use a power series to approximate the definite integral to six deci...
 8.8.1.28: Determine whether the sequence converges or diverges. If it converg...
 8.8.8.28: The period of a pendulum with length that makes a maximum angle wit...
 8.8.2.28: Find the values of for which the series converges. Find the sum of ...
 8.28: Find the sum of the series.n0 1 n n 32n 2n!
 8.8.3.29: Let be the sum of a series that has been shown to be convergent by ...
 8.8.4.29: Determine whether the series is absolutely convergent, conditionall...
 8.8.5.29: Suppose the series has radius of convergence 2 and the series has r...
 8.8.6.29: Use a power series to approximate the definite integral to six deci...
 8.8.1.29: If $1000 is invested at 6% interest, compounded annually, then afte...
 8.8.8.29: In Section 3.6 we considered Newtons method for approximating a roo...
 8.8.2.29: Find the values of for which the series converges. Find the sum of ...
 8.29: Find the sum of the series.1 e e 2 2! e 3 3! e 4 4!
 8.8.3.30: (a) Find the partial sum of the series . Use Exercise 29(a) to esti...
 8.8.4.30: Determine whether the series is absolutely convergent, conditionall...
 8.8.5.30: Suppose that the radius of convergence of the power series is . Wha...
 8.8.6.30: Use a power series to approximate the definite integral to six deci...
 8.8.1.30: Find the first 40 terms of the sequence defined by and . Do the sam...
 8.8.8.30: Use the following outline to prove that is an irrational number. (a...
 8.8.2.30: We have seen that the harmonic series is a divergent series whose t...
 8.30: Express the repeating decimal as a fraction.
 8.8.3.31: (a) Use the sum of the first 10 terms and Exercise 29(a) to an esti...
 8.8.4.31: Determine whether the series is absolutely convergent, conditionall...
 8.8.6.31: Use the result of Example 6 to compute correct to five decimal places.
 8.8.7.31: Use a Maclaurin series derived in this section to obtain the Maclau...
 8.8.1.31: Suppose you know that is a decreasing sequence and all its terms li...
 8.8.2.31: If the partial sum of a series is find and .
 8.31: Show that for all .
 8.8.3.32: Find the sum of the series correct to three decimal places.
 8.8.4.32: Determine whether the series is absolutely convergent, conditionall...
 8.8.6.32: Show that the function is a solution of the differential equation
 8.8.7.32: Use a Maclaurin series derived in this section to obtain the Maclau...
 8.8.1.32: (a) If is convergent, show that (b) A sequence is defined by and fo...
 8.8.2.32: If the partial sum of a series is , find and .
 8.32: For what values of does the series converge?
 8.8.3.33: (a) Use a graph of to show that if is the partial sum of the harmon...
 8.8.4.33: Determine whether the series is absolutely convergent, conditionall...
 8.8.6.33: (a) Show that (the Bessel function of order 0 given in Example 4) s...
 8.8.7.33: Use a Maclaurin series derived in this section to obtain the Maclau...
 8.8.1.33: Determine whether the sequence is increasing, decreasing, or not mo...
 8.8.2.33: When money is spent on goods and services, those that receive the m...
 8.33: Find the sum of the series correct to four decimal places.
 8.8.3.34: Show that if we want to approximate the sum of the series so that t...
 8.8.4.34: Determine whether the series is absolutely convergent, conditionall...
 8.8.6.34: The Bessel function of order 1 is defined by (a) Show that satisfie...
 8.8.7.34: Use a Maclaurin series derived in this section to obtain the Maclau...
 8.8.1.34: Determine whether the sequence is increasing, decreasing, or not mo...
 8.8.2.34: A certain ball has the property that each time it falls from a heig...
 8.34: (a) Show that the series is convergent. (b) Deduce that .
 8.8.3.35: The meaning of the decimal representation of a number (where the di...
 8.8.4.35: Determine whether the series is absolutely convergent, conditionall...
 8.8.6.35: (a) Show that the function is a solution of the differential equati...
 8.8.7.35: Use a Maclaurin series derived in this section to obtain the Maclau...
 8.8.1.35: Determine whether the sequence is increasing, decreasing, or not mo...
 8.8.2.35: What is the value of if ?
 8.35: Prove that if the series is absolutely convergent, then the series ...
 8.8.3.36: Show that if and is convergent, then ln1 an is convergent. an
 8.8.4.36: Determine whether the series is absolutely convergent, conditionall...
 8.8.6.36: Let . Show that the series converges for all values of but the seri...
 8.8.7.36: Use a Maclaurin series derived in this section to obtain the Maclau...
 8.8.1.36: Determine whether the sequence is increasing, decreasing, or not mo...
 8.8.2.36: Graph the curves , , for on a common screen. By finding the areas b...
 8.36: Find the radius of convergence and interval of convergence of the s...
 8.8.3.37: If is a convergent series with positive terms, is it true that is a...
 8.8.4.37: Determine whether the series is absolutely convergent, conditionall...
 8.8.6.37: Let Find the intervals of convergence for , , and .
 8.8.7.37: Find the Maclaurin series of (by any method) and its radius of conv...
 8.8.1.37: Find the limit of the sequence s2 , s2s2 , s2s2s2 ,...}
 8.8.2.37: The figure shows two circles and of radius 1 that touch at . is a c...
 8.37: Find the radius of convergence and interval of convergence of the s...
 8.8.3.38: (a) Suppose that and are series with positive terms and is converge...
 8.8.4.38: Determine whether the series is absolutely convergent, conditionall...
 8.8.6.38: (a) Starting with the geometric series , find the sum of the series...
 8.8.7.38: Find the Maclaurin series of (by any method) and its radius of conv...
 8.8.1.38: A sequence is given by , . (a) By induction or otherwise, show that...
 8.8.2.38: A right triangle is given with and . is drawn perpendicular to , is...
 8.38: Find the radius of convergence and interval of convergence of the s...
 8.8.3.39: (a) Suppose that and are series with positive terms and is divergen...
 8.8.4.39: For which of the following series is the Ratio Test inconclusive (t...
 8.8.6.39: Use the power series for to prove the following expression for as t...
 8.8.7.39: Use the Maclaurin series for to calculate correct to five decimal p...
 8.8.1.39: Use induction to show that the sequence defined by , is increasing ...
 8.8.2.39: What is wrong with the following calculation? (Guido Ubaldus though...
 8.39: Find the radius of convergence and interval of convergence of the s...
 8.8.3.40: Give an example of a pair of series and with positive terms where a...
 8.8.4.40: For which positive integers is the following series convergent?
 8.8.7.40: Use the Maclaurin series for to compute correct to five decimal pla...
 8.8.1.40: Show that the sequence defined by satisfies and is decreasing. Dedu...
 8.8.2.40: Suppose that is known to be a convergent series. Prove that is a di...
 8.40: Find the radius of convergence of the series n1 2n! n! 2 x n
 8.8.3.41: Prove that if and converges, then also converges.
 8.8.4.41: (a) Show that converges for all . (b) Deduce that for all .
 8.8.7.41: a) Use the binomial series to expand . (b) Use part (a) to find the...
 8.8.1.41: (a) Fibonacci posed the following problem: Suppose that rabbits liv...
 8.8.2.41: Prove part (i) of Theorem 8.
 8.41: Find the Taylor series of at .
 8.8.3.42: Find all positive values of for which the series converges.
 8.8.4.42: Around 1910, the Indian mathematician Srinivasa Ramanujan discovere...
 8.8.7.42: (a) Expand as a power series. (b) Use part (a) to estimate correct ...
 8.8.1.42: (a) Let , , ,..., , where is a continuous function. If , show that ...
 8.8.2.42: If is divergent and , show that is divergent.
 8.42: Find the Taylor series of at .
 8.8.4.43: Prove the Root Test. [Hint for part (i): Take any number such that ...
 8.8.7.43: Evaluate the indefinite integral as an infinite series.y x cosx dx ...
 8.8.1.43: We know that [from (8) with ]. Use logarithms to determine how larg...
 8.8.2.43: If is convergent and is divergent, show that the series is divergen...
 8.43: Find the Maclaurin series for and its radius of convergence. You ma...
 8.8.7.44: Evaluate the indefinite integral as an infinite series. sin x x y x...
 8.8.1.44: Use Definition 2 directly to prove that when .
 8.8.2.44: If and are both divergent, is necessarily divergent?
 8.44: Find the Maclaurin series for and its radius of convergence. You ma...
 8.8.7.45: Evaluate the indefinite integral as an infinite series.sx dx 3 1 dx
 8.8.1.45: Prove Theorem 6. [Hint: Use either Definition 2 or the Squeeze Theo...
 8.8.2.45: Suppose that a series has positive terms and its partial sums satis...
 8.45: Find the Maclaurin series for and its radius of convergence. You ma...
 8.46: Find the Maclaurin series for and its radius of convergence. You ma...
 8.8.7.46: Evaluate the indefinite integral as an infinite series.y ex 1 x y s...
 8.8.1.46: (a) Show that if and , then is convergent and . (b) If and find the...
 8.8.2.46: The Fibonacci sequence was defined in Section 8.1 by the equations ...
 8.47: Find the Maclaurin series for and its radius of convergence. You ma...
 8.8.7.47: Use series to approximate the definite integral to within the indic...
 8.8.1.47: The size of an undisturbed fish population has been modeled by the ...
 8.8.2.47: The Cantor set, named after the German mathematician Georg Cantor (...
 8.48: Find the Maclaurin series for and its radius of convergence. You ma...
 8.8.7.48: Use series to approximate the definite integral to within the indic...
 8.8.2.48: (a) A sequence is defined recursively by the equation for , where a...
 8.49: Find the Maclaurin series for and its radius of convergence. You ma...
 8.8.7.49: Use series to approximate the definite integral to within the indic...
 8.8.2.49: Consider the series (a) Find the partial sums and . Do you recogniz...
 8.50: Find the Maclaurin series for and its radius of convergence. You ma...
 8.8.7.50: Use series to approximate the definite integral to within the indic...
 8.8.2.50: In the figure there are infinitely many circles approaching the ver...
 8.51: Evaluate as an infinite series.
 8.8.7.51: Use series to evaluate the limit.lim xl0 x tan1 x x 3 y
 8.52: Use series to approximate correct to two decimal places.
 8.8.7.52: Use series to evaluate the limit. lim xl0 1 cos x 1 x e x lim
 8.53: (a) Approximate by a Taylor polynomial with degree at the number . ...
 8.8.7.53: Use series to evaluate the limit.lim xl0 sin x x 1 6 x 3 x 5 53
 8.54: (a) Approximate by a Taylor polynomial with degree at the number . ...
 8.8.7.54: Use the series in Example 11(b) to evaluate We found this limit in ...
 8.55: Use series to evaluate the following limit. lim xl0 sin x x x 3 f
 8.8.7.55: Use multiplication or division of power series to find the first th...
 8.56: The force due to gravity on an object with mass at a height above t...
 8.8.7.56: Use multiplication or division of power series to find the first th...
 8.57: Suppose that for all . (a) If is an odd function, show that (b) If ...
 8.8.7.57: Use multiplication or division of power series to find the first th...
 8.58: If , show that .
 8.8.7.58: Use multiplication or division of power series to find the first th...
 8.8.7.59: Find the sum of the series.n0 1 n x 4n n! 59.
 8.8.7.60: Find the sum of the series.n0 1 n 2n 62n 2n! n
 8.8.7.61: Find the sum of the series.n0 1 n 2n1 42n1 2n 1! n0
 8.8.7.62: Find the sum of the series.n0 3n 5n n!
 8.8.7.63: Find the sum of the series.3 9 2! 27 3! 81 4!
 8.8.7.64: Find the sum of the series.1 ln 2 ln 2 2 2! ln 2 3 3! 3 9
 8.8.7.65: (a) Expand as a power series. (b) Use part (a) to find the sum of t...
 8.8.7.66: (a) Expand as a power series. (b) Use part (a) to find the sum of t...
 8.8.7.67: a) Use the binomial series to find the Maclaurin series of . (b) Us...
 8.8.7.68: (a) Use the binomial series to find the Maclaurin series of . (b) U...
 8.8.7.69: Use the following steps to prove (18). (a) Let . Differentiate this...
 8.8.7.70: (a) Show that the function defined by is not equal to its Maclaurin...
Solutions for Chapter 8: SERIES
Full solutions for Essential Calculus (Available Titles CengageNOW)  1st Edition
ISBN: 9780495014423
Solutions for Chapter 8: SERIES
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Essential Calculus (Available Titles CengageNOW) was written by and is associated to the ISBN: 9780495014423. This textbook survival guide was created for the textbook: Essential Calculus (Available Titles CengageNOW), edition: 1. Chapter 8: SERIES includes 385 full stepbystep solutions. Since 385 problems in chapter 8: SERIES have been answered, more than 17214 students have viewed full stepbystep solutions from this chapter.

Absolute value of a vector
See Magnitude of a vector.

Angle of depression
The acute angle formed by the line of sight (downward) and the horizontal

Central angle
An angle whose vertex is the center of a circle

Compounded continuously
Interest compounded using the formula A = Pert

Constant of variation
See Power function.

Difference of functions
(ƒ  g)(x) = ƒ(x)  g(x)

Directrix of a parabola, ellipse, or hyperbola
A line used to determine the conic

DMS measure
The measure of an angle in degrees, minutes, and seconds

Higherdegree polynomial function
A polynomial function whose degree is ? 3

Independent variable
Variable representing the domain value of a function (usually x).

Instantaneous rate of change
See Derivative at x = a.

Measure of an angle
The number of degrees or radians in an angle

Measure of center
A measure of the typical, middle, or average value for a data set

Onetoone rule of exponents
x = y if and only if bx = by.

Probability of an event in a finite sample space of equally likely outcomes
The number of outcomes in the event divided by the number of outcomes in the sample space.

Semiperimeter of a triangle
Onehalf of the sum of the lengths of the sides of a triangle.

Speed
The magnitude of the velocity vector, given by distance/time.

Symmetric about the yaxis
A graph in which (x, y) is on the graph whenever (x, y) is; or a graph in which (r, ?) or (r, ?, ?) is on the graph whenever (r, ?) is

Tangent
The function y = tan x

Weights
See Weighted mean.