?Suppose that \(f(x)=\sum_{n=0}^{\infty} c_{n} x^{n}\) for all x.(a) If f is an odd

(a) What is a convergent sequence? (b) What is a convergent series? (c) What does mean? (d) What does mean?

(a) What is a bounded sequence? (b) What is a monotonic sequence? (c) What can you say about a bounded monotonic sequence?

(a) What is a geometric series? Under what circumstances is it convergent? What is its sum? (b) What is a -series? Under what circumstances is it convergent?

Suppose and is the th partial sum of the series. What is What is

State the following. (a) The Test for Divergence (b) The Integral Test (c) The Direct Comparison Test (d) The Limit Comparison Test (e) The Alternating Series Test (f) The Ratio Test (g) The Root Test

(a) What is an absolutely convergent series? (b) What can you say about such a series? (c) What is a conditionally convergent series?

(a) If a series is convergent by the Integral Test, how do you estimate its sum? (b) If a series is convergent by the Direct Comparison Test, how do you estimate its sum? (c) If a series is convergent by the Alternating Series Test, how do you estimate its sum?

(a) Write the general form of a power series. (b) What is the radius of convergence of a power series? (c) What is the interval of convergence of a power series?

Suppose \(f(x)\) is the sum of a power series with radius of convergence \(R.\) (a) How do you differentiate \(f\)? What is the radius of convergence of the series for \(f^{\prime}\) ? (b) How do you integrate \(f\)? What is the radius of convergence of the series for \(\int f(x) d x\)? Equation Transcription: Text Transcription: f(x) R f f’ f int f(x) dx

(a) Write an expression for the nth-degree Taylor polynomial of f centered at a. (b) Write an expression for the Taylor series of f centered at a. (c) Write an expression for the Maclaurin series of f. (d) How do you show that \(f(x)\) is equal to the sum of its Taylor series? (e) State Taylor's Inequality.

Write the Maclaurin series and the interval of convergence for each of the following functions. (a) \(1 /(1-x)\) (b) \(e^{x}\) (c) \(\sin x\) (d) \(\cos x\) (e) \(\tan ^{-1} x\) (f) \(\ln (1+x)\) Equation Transcription: Text Transcription: 1/(1-x) e^x sin x cos x tan^-1x ln(1+x)

Write the binomial series expansion of . What is the radius of convergence of this series?

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If , then is convergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The series \(\sum_{n=1}^{\infty} n^{-\sin 1}\) is convergent. Equation Transcription: Text Transcription: sum^n=1 _infty n^-sin 1

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(\lim _{n \rightarrow \infty} a_{n}=L\), then \(\lim _{n \rightarrow \infty} a_{2 n+1}=L\) Equation Transcription: Text Transcription: lim _{n rightarrow infty} a_{n} = L lim _{n rightarrow infty} a_{2 n+1} = L

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If is convergent, then is convergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If is convergent, then is convergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(\Sigma c_{n} x^{n}\) diverges when \(x=6\), then it diverges when \(x=10\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The Ratio Test can be used to determine whether \(\Sigma 1 / n^{3}\) converges.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The Ratio Test can be used to determine whether converges.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If and diverges, then diverges.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{n}{n^{3}+1}\) Equation Transcription: Text Transcription: sum_n=1^infinity n/n^3+1

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(\Sigma a_{n}\) is divergent, then \(\Sigma\left|a_{n}\right|\) is divergent.

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{n^{3}}{5^{n}}\) Equation Transcription: Text Transcription: sum_n=1^infinity n^3/5^n

A sequence \(\left\{a_{n}\right\}\) is defined recursively by the equations \(a_{0}=a_{1}=1 \quad n(n-1) a_{n}=(n-1)(n-2) a_{n-1}-(n-3) a_{n-2}\) Find the sum of the series \(\sum_{n=0}^{\infty} a_{n}\). ________________ Equation Transcription: Text Transcription: {a_n} a_{0}=a_{1}=1 \quad n(n-1) a_{n}=(n-1)(n-2) a_{n-1}-(n-3) a_{n-2} sum_n=0^infinity a_n

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If and are divergent, then is divergent.

Starting with the vertices \(P_{1}(0,1), P_{2}(1,1), P_{3}(1,0), P_{4}(0,0)\) of a square, we construct further points as shown in the figure: \(P_{5}\) is the midpoint of \(P_{1} P_{2}, P_{6}\) is the midpoint of \(P_{2} P_{3}, P_{7}\) is the midpoint of \(P_{3} P_{4}\), and so on. The polygonal spiral path \(P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} P_{7} \ldots\) approaches a point \(P\) inside the square. (a) If the coordinates of \(P_{n}\) are \(\left(x_{n}, y_{n}\right)\), show that \(\frac{1}{2} x_{n}+x_{n+1}+x_{n+2}+x_{n+3}=2\) and find a similar equation for the \(y\)-coordinates. (b) Find the coordinates of \(P\). Equation Transcription: Text Transcription: P_1(0,1),P_2(1,1),P_3(1,0),P_4(0,0) P_5 P_3P_4 P_1P_2P_3P_4P_5P_6P_7... 1/2 x_n+x_n+1+x_n+2+x_n+3=2

Find the sum of the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2 n+1) 3^{n}}\). Equation Transcription: Text Transcription: sum_n=1^infinity (-1)^n/(2n+1)3^n

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(a_{n}>0\) and \(\lim _{n \rightarrow \infty}\left(a_{n+1} / a_{n}\right)<1\), then \(\lim _{n \rightarrow \infty} a_{n}=0\).

Find all the solutions of the equation \(1+\frac{x}{2 !}+\frac{x^{2}}{4 !}+\frac{x^{3}}{6 !}+\frac{x^{4}}{8 !}+\cdots=0\) [Hint: Consider the cases \(x \geqslant 0\) and \(x<0\) separately. Equation Transcription: ?0 Text Transcription: 1+\frac{x}{2 !}+\frac{x^{2}}{4 !}+\frac{x^{3}}{6 !}+\frac{x^{4}}{8 !}+\cdots=0 x \geqslant 0 x<0

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(\lim _{n \rightarrow \infty} a_{n}=2\), then \(\lim _{n \rightarrow \infty}\left(a_{n+3}-a_{n}\right)=0\).

Consider the series whose terms are the reciprocals of the positive integers that can be written in base 10 notation without using the digit Show that this series is convergent and the sum is less than 90 .

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{\sqrt{n+1}-\sqrt{n-1}}{n}\) Equation Transcription: Text Transcription: sum_n=1^infinity \frac{\sqrt{n+1}-\sqrt{n-1}}{n}

Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.

Let \(\left\{P_{n}\right\}\) be a sequence of points determined as in the figure. Thus \(\left|A P_{1}\right|=1\), \(\left|P_{n} P_{n+1}\right|=2^{n-1}\), and angle \(A P_{n} P_{n+1}\) is a right angle. Find \(\lim _{n \rightarrow \infty} \angle P_{n} A P_{n+1}\). Equation Transcription: , ? Text Transcription: {P_n} |AP_1|=1, |P_n P_n+1|=2^n-1 AP_n P_n+1 lim_n right arrow pm angle P_n AP_n+1

Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. \(a_{n}=\frac{n^{3}}{1+n^{2}}\)

Find the sum of the series \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{12}+\cdots\) where the terms are the reciprocals of the positive integers whose only prime factors are \(2 \mathrm{~s}\) and \(3 \mathrm{~s}\). Equation Transcription: Text Transcription: 1+1/2+1/3+1/4+1/6+1/8+1/9+1/12+ dots 2 s 23s

Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. \(a_{n}=\frac{n \sin n}{n^{2}+1}\) Equation Transcription: Text Transcription: a_n= nsinn/n^2+1

Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. \(a_{n}=\frac{\ln n}{\sqrt{n}}\) Equation Transcription: Text Transcription: a_n= lnn/sqrt n

Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.

Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.

A sequence is defined recursively by the equations \(a_{1}=1\), \(a_{n+1}=\frac{1}{3}\left(a_{n}+4\right)\). Show that \(\left\{a_{n}\right\}\) is increasing and \(a_{n}<2\) for all n. Deduce that \(\left\{a_{n}\right\}\) is convergent and find its limit.

Suppose you have a large supply of books, all the same size, and you stack them at the edge of a table, with each book extending farther beyond the edge of the table than the one beneath it. Show that it is possible to do this so that the top book extends entirely beyond the table. In fact, show that the top book can extend any distance at all beyond the edge of the table if the stack is high enough. Use the following method of stacking: The top book extends half its length beyond the second book. The second book extends a quarter of its length beyond the third. The third extends one-sixth of its length beyond the fourth, and so on. (Try it yourself with a deck of cards.) Consider centers of mass.

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{n^{2}+1}{n^{3}+1}\) Equation Transcription: Text Transcription: sum_{n=1}^{infty} frac{n^{2}+1}{n^{3}+1}

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{1}{n \sqrt{\ln n}}\) Equation Transcription: Text Transcription: sum over n=2 infty 1/n sqrt ln n

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \ln \left(\frac{n}{3 n+1}\right)\) Equation Transcription: Text Transcription: sum_n=1^infinity ln(n/3n+1)

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{\cos 3 n}{1+(1.2)^{n}}\) Equation Transcription: Text Transcription: sum_n=1^infinity cos 2n/1+(1.2)^n

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{n^{2 n}}{\left(1+2 n^{2}\right)^{n}}\) Equation Transcription: Text Transcription: sum_n=1^infinity frac{n^2n}{(1+2n^2)^n}

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{(-5)^{2 n}}{n^{2} 9^{n}}\) Equation Transcription: Text Transcription: sum_{n=1}^{infty} frac{(-5)^2n}{n^{2} 9^{n}}

Determine whether the series is convergent or divergent.

Determine whether the series is convergent or divergent.

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty}(-1)^{n-1} n^{-1 / 3}\)

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty}(-1)^{n-1} n^{-3}\)

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(-1)^{n}(n+1) 3^{n}}{2^{2 n+1}}\) Equation Transcription: Text Transcription: sum_n=1^infinity frac{(-1)^{n}(n+1)3^n}{2^2n+1}

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=2}^{\infty} \frac{(-1)^{n} \sqrt{n}}{\ln n}\) Equation Transcription: Text Transcription: sum_n=1^infinity (-1)^n sqrt n/ln n

Find the sum of the series. \(\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{2^{3 n}}\) Equation Transcription: Text Transcription: sum_n=1^infinity {(-3)^n-1}/{2^3n}

Find the sum of the series. \(\sum_{n=1}^{\infty} \frac{1}{n(n+3)}\) Equation Transcription: Text Transcription: sum_n=1^infinity 1/n(n+3)

Find the sum of the series. \(\sum_{n=1}^{\infty}\left[\tan ^{-1}(n+1)-\tan ^{-1} n\right]\) Equation Transcription: Text Transcription: sum_n=1^infinity [tan^-1(n+1)-tan^-1n]

Find the sum of the series. \(\sum_{n=0}^{\infty} \frac{(-1)^{n} \pi^{n}}{3^{2 n}(2 n) !}\)

Find the sum of the series. \(1-e+\frac{e^{2}}{2 !}-\frac{e^{3}}{3 !}+\frac{e^{4}}{4 !}-\cdots\) Equation Transcription: Text Transcription: 1-e+\frac{e^{2}}{2 !}-\frac{e^{3}}{3 !}+\frac{e^{4}}{4 !}-dots

Express the repeating decimal \(4.17326326326 \ldots\) as a fraction. Equation Transcription: Text Transcription: 4.17326326326. . .

Show that \(\cosh x \geqslant 1+\frac{1}{2} x^{2}\) for all \(x\). Equation Transcription: ? Text Transcription: cosh geqslant 1+ 1/2 x^2 x

For what values of x does the series \(\sum_{n=1}^{\infty}(\ln \ x)^{n}\) converge?

Find the sum of the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{5}}\) correct to four decimal places. Equation Transcription: Text Transcription: sum_n=1^infinity (-1)^n+1/n^5

(a) Find the partial sum \(s_{5}\) of the series \(\sum_{n-1}^{\infty} 1 / n^{6}\) and estimate the error in using it as an approximation to the sum of the series. (b) Find the sum of this series correct to five decimal places. Equation Transcription: Text Transcription: s_5 sum_n=1^infinity 1/n^6

Use the sum of the first eight terms to approximate the sum of the series \(\sum_{n-1}^{\infty}\left(2+5^{n}\right)^{-1}\). Estimate the error involved in this approximation.

(a) Show that the series \(\sum_{n=1}^{\infty} \frac{n^{n}}{(2 n) !}\) is convergent. (b) Deduce that \(\lim _{n \rightarrow \infty} \frac{n^{n}}{(2 n) !}=0\). Equation Transcription: Text Transcription: sum_n=1^infinity n^n/(2n)! lim_n right arrow infinity n^n/(2n)!=0

Prove that if the series \(\sum_{n-1}^{\infty} a_{n}\) is absolutely convergent, then the series \(\sum_{n=1}^{\infty}\left(\frac{n+1}{n}\right) a_{n}\) is also absolutely convergent. Equation Transcription: Text Transcription: sum_n-1^infinity a_n sum_n=1^infinity (n+1/n)a_n

Find the radius of convergence and interval of convergence of the series. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{x^{n}}{n^{2} 5^{n}}\)

Find the radius of convergence and interval of convergence of the series. \(\sum_{n=1}^{\infty} \frac{(x+2)^{n}}{n 4^{n}}\) Equation Transcription: Text Transcription: sum_n=1 ^infinity (x+2)^n /n4^n

Find the radius of convergence and interval of convergence of the series. \(\sum_{n=1}^{\infty} \frac{2^{n}(x-2)^{n}}{(n+2) !}\) Equation Transcription: Text Transcription: sum_n=1 ^infinity 2^n (x-2)^n /(n+2)!

Find the radius of convergence and interval of convergence of the series. \(\sum_{n=0}^{\infty} \frac{2^{n}(x-3)^{n}}{\sqrt{n+3}}\) Equation Transcription: Text Transcription: sum_n=0 ^infinity 2^n (x-3)^n /sqrt n+3

Find the radius of convergence of the series \(\sum_{n=1}^{\infty} \frac{(2 n) !}{(n !)^{2}} x^{n}\) Equation Transcription: Text Transcription: sum_n=1 ^infinity (2n)! /(n!)^2 X^n

Find the Taylor series of\(f(x)=\sin x\) at \(a=\pi / 6\). Equation Transcription: Text Transcription: f(x)=sin x a=pi/6

Find the Taylor series of \(f(x)=\cos x\) at \(a=\pi / 3\). Equation Transcription: Text Transcription: f(x)=cos x a=pi/3

Find the Maclaurin series for \(f\) and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1. \(f(x)=\frac{x^{2}}{1+x}\) Equation Transcription: Text Transcription: f f(x)=x^2/1+x

Find the Maclaurin series for \(f\) and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1. \(f(x)=\tan ^{-1}\left(x^{2}\right)\) Equation Transcription: Text Transcription: f f(x)=tan^-1(x^2)

Find the Maclaurin series for \(f\) and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1. \(f(x)=\ln (4-x)\) Equation Transcription: Text Transcription: f f(x)=kn(4-x)

Find the Maclaurin series for \(f\) and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1. \(f(x)=x e^{2 x}\) Equation Transcription: Text Transcription: f f(x)=xe^2x

Find the Maclaurin series for \(f\) and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1. \(f(x)=\sin \left(x^{4}\right)\) Equation Transcription: Text Transcription: f f(x)=sin(x^4)

Find the Maclaurin series for f and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1. \(f(x)=10^{x}\)

Find the Maclaurin series for \(f\) and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1. \(f(x)=1 / \sqrt[4]{16-x}\) Equation Transcription: Text Transcription: f f(x)=1/4 th root 16?x

Find the Maclaurin series for \(f\) and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1. \(f(x)=(1-3 x)^{-5}\) Equation Transcription: Text Transcription: f f(x)=(1-3x)^-5

Evaluate \(\int \frac{e^{x}}{x} d x\) as an infinite series. ________________ Equation Transcription: Text Transcription: integral e^x/x dx

Use series to approximate \(\int_{0}^{1} \sqrt{1+x^{4}} d x\) correct to two decimal places. Equation Transcription: Text Transcription: integral_0^1 square root 1+x^4 dx ________________

(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number \(a\). (b) Graph \(f\) and \(T_{n}\) on a common screen. (c) Use Taylor's Inequality to estimate the accuracy of the approximation \(f(x) \approx T_{n}(x)\) when \(x\) lies in the given interval. (d) Check your result in part (c) by graphing \(\left|R_{n}(x)\right|\). \(f(x)=\sqrt{x}, \quad a=1, \quad n=3, \quad 0.9 \leqslant x \leqslant 1.1\) Equation Transcription: f(x) ? Tn(x) 0.9 ? ? 1.1 Text Transcription: f n a Tn f(x) ? Tn(x) x Rn(x) f(x)=x, a=1, n=3, 0.9 ? x ? 1.1

(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number \(a\). (b) Graph \(f\) and \(T_{n}\) on a common screen. (c) Use Taylor's Inequality to estimate the accuracy of the approximation \(f(x) \approx T_{n}(x)\) when \(x\) lies in the given interval. (d)Check your result in part (c) by graphing \(\left|R_{n}(x)\right|\). \(f(x)=\sec x, \quad a=0, \quad n=2, \quad 0 \leqslant x \leqslant \pi / 6\) Equation Transcription: ? ? Text Transcription: f n a f T_n f(x)approx T_n(x) x |R_n(x)| f(x)sec x, a=0, n=2, 0 leqslant x leqslant pi/6

Use series to evaluate the following limit. \(\lim _{x \rightarrow 0} \frac{\sin x-x}{x^{3}}\) Equation Transcription: Text Transcription: lim over x rightarrow 0 sin x-x/x^3

The force due to gravity on an object with mass \(m\) at a height \(h\) above the surface of the earth is \(F=\frac{m g R^{2}}{(R+h)^{2}}\) where \(R\) is the radius of the earth and \(g\) is the acceleration due to gravity for an object on the surface of the earth. (a) Express \(F\) as a series in powers of \(h / R\). (b) Observe that if we approximate \(F\) by the first term in the series, we get the expression \(F \approx m g\) that is usually used when \(h\) is much smaller than \(R\). Use the Alternating Series Estimation Theorem to estimate the range of values of \(h\) for which the approximation \(F \approx m g\) is accurate to within one percent. (Use \(R=6400 \mathrm{~km} .)\) Equation Transcription: Text Transcription: m h F=mgR^2/(R+h)^2 F h/R F approx mg R= 6400 km

Suppose that \(f(x)=\sum_{n=0}^{\infty} c_{n} x^{n}\) for all x. (a) If f is an odd function, show that \(c_{0}=c_{2}=c_{4}=\ldots=0\) (b) If f is an even function, show that \(c_{1}=c_{3}=c_{5}=\ldots=0\) Equation Transcription: Text Transcription: f(x)=sum_n=0 ^infinity c_n X^n c_0=c_2=c_4=...=0 c_1=c_3=c_5=...=0

If \(f(x)=e^{x^{2}} \text {, show that } f^{(2 n)}(0)=\frac{(2 n) !}{n !}\). Equation Transcription: Text Transcription: f(x)=e^x^2 f^(2n)(0)=(2n)/!n!

(a) Show that \(\tan \frac{1}{2} x=\cot \frac{1}{2} x-2 \cot x\). (b) Find the sum of the series \(\sum_{n=1}^{\infty} \frac{1}{2^{n}} \tan \frac{x}{2^{n}}\) Equation Transcription: Text Transcription: tan 1/2 x= cot 1/2 x- 2 cot x Sum_n=1^infinity 1/2^n tan x/2^n

Let be a sequence of points determined as in the figure. Thus , , and angle is a right angle. Find .

To construct the snowflake curve, start with an equilateral triangle with sides of length 1. Step 1 in the construction is to divide each side into three equal parts, construct an equi- lateral triangle on the middle part, and then delete the middle part (see the figure). Step 2 is to repeat step 1 for each side of the resulting polygon. This process is repeated at each succeeding step. The snowflake curve is the curve that results from repeating this process indefinitely. (a) Let , and represent the number of sides, the length of a side, and the total length of the nth approximating curve (the curve obtained after step of the construction), respectively. Find formulas for , and . (b) Show that as . (c) Sum an infinite series to find the area enclosed by the snowflake curve. Note: Parts (b) and (c) show that the snowflake curve is infinitely long but encloses only a finite area.

Find the sum of the series \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{12}+\cdots\) where the terms are the reciprocals of the positive integers whose only prime factors are \(2 \mathrm{~s}\) and \(3 \mathrm{~s}\).

(a) Show that for \(x y \neq-1\), \(arctan x-\arctan y=\arctan \frac{x-y}{1+x y}\) if the left side lies between \(-\pi / 2 and \pi / 2\). (b) Show that \(arctan \arctan \frac{120}{119}-\arctan \frac{1}{239}=\pi / 4\). (c) Deduce the following formula of John Machin (1680–1751): \(4 \arctan \frac{1}{5}-\arctan \frac{1}{239}=\frac{\pi}{4}\) (d) Use the Maclaurin series for arctan to show that \(0.1973955597<\arctan \frac{1}{5}<0.1973955616\) (e) Show that \(0.004184 .75<\arctan \frac{1}{239}<0.004184077\) (f) Deduce that, correct to seven decimal places, \(\pi \approx 3.1415927\). Machin used this method in 1706 to find \(\pi\) correct to 100 decimal places. Recently, with the aid of computers, the value of \(\pi\) has been computed to increasingly greater accuracy, well into the trillions of decimal places. Equation Transcription: Text Transcription: xy neq -1 arctan x-arctan y=arctan x-y/1+xy -pi/2 pi/2 arctan 120/119- arctan 1/239=pi/4 4 arctan 1/5-arctan 1/239=pi/4 0.1973955597<arctan 1/5 < 0.1973955616 0.004184.75<arctan 1/239<0.004184077 pi approx 3.1415927 pi pi

(a) Prove a formula similar to the one in Problem 5(a) but involving arccot instead of arctan. (b) Find the sum of the series \(\sum_{n=0}^{\infty} \operatorname{arccot}\left(n^{2}+n+1\right)\).

Use the result of Problem 5 (a) to find the sum of the series \(\sum_{n=1}^{\infty} \arctan \left(2 / n^2\right)\).

If \(a_0+a_1+a_2+\cdots+a_k=0\), show that \(\lim _{n \rightarrow \infty}\left(a_0 \sqrt{n}+a_1 \sqrt{n+1}+a_2 \sqrt{n+2}+\cdots+a_k \sqrt{n+k}\right)=0\) If you don't see how to prove this, try the problem-solving strategy of using analogy. Try the special cases k=1 and k=2 first. If you can see how to prove the assertion for these cases, then you will probably see how to prove it in general.

Find the interval of convergence of and find its sum.

Suppose you have a large supply of books, all the same size, and you stack them at the edge of a table, with each book extending farther beyond the edge of the table than the one beneath it. Show that it is possible to do this so that the top book extends entirely beyond the table. In fact, show that the top book can extend any distance at all beyond the edge of the table if the stack is high enough. Use the following method of stacking: The top book extends half its length beyond the second book. The second book extends a quarter of its length beyond the third. The third extends one-sixth of its length beyond the fourth, and so on. (Try it yourself with a deck of cards.) Consider centers of mass.

Find the sum of the series \(\sum_{n=2}^{\infty} \ln \left(1-\frac{1}{n^{2}}\right)\). Equation Transcription: Text Transcription: sum_n=2^infinity ln(1-1/n^2)

If \(p>1\), evaluate the expression \(\frac{1+\frac{1}{2^{p}}+\frac{1}{3^{p}}+\frac{1}{4^{p}}+\cdots}{1-\frac{1}{2^{p}}+\frac{1}{3^{p}}-\frac{1}{4^{p}}+\cdots}\)

Suppose that circles of equal diameter are packed tightly in n rows inside an equilateral triangle. (The figure illustrates the case \(n=4\). ) If A is the area of the triangle and \(A_{n}\) is the total area occupied by the n rows of circles, show that \(\lim _{n \rightarrow \infty} \frac{A_{n}}{A}=\frac{\pi}{2 \sqrt{3}}\) Equation Transcription: Text Transcription: n=4 A_n lim _n right arrow infinity A_n /A = pi/2sqrt3

A sequence \(\left\{a_{n}\right\}\) is defined recursively by the equations \(a_{0}=a_{1}=1 \quad n(n-1) a_{n}=(n-1)(n-2) a_{n-1}-(n-3) a_{n-2}\) Find the sum of the series \(\sum_{n=0}^{\infty} a_{n}\).

If the curve \(y=e^{-x / 10} \sin \ x, \ x \geqslant 0\), is rotated about the x-axis the resulting solid looks like an infinite decreasing string of beads. (a) Find the exact volume of the nth bead. (Use either a table of integrals or a computer algebra system.) (b) Find the total volume of the beads.

Starting with the vertices of a square, we construct further points as shown in the figure: is the midpoint of is the midpoint of is the midpoint of , and so on. The polygonal spiral path approaches a point inside the square. (a) If the coordinates of are , show that and find a similar equation for the -coordinates. (b) Find the coordinates of .

Find the sum of the series .

Carry out the following steps to show that \(\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.2}+\frac{1}{7.8}+\ldots=\ln 2\) (a) Use the formula for the sum of a finite geometric series (11.2.3) to get an expression for \(1-x+x^{2}-x^{3}+\ldots+x^{2 n-2}-x^{2 n-1}\) (b) Integrate the result of part (a) from 0 to 1 to get an expression for \(1-\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots+\frac{1}{2 n-1}-\frac{1}{2 n}\) as an integral. (c) Deduce from part (b) that \(\left|\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+\ldots+\frac{1}{(2 n-1)(2 n)}-\int_{0}^{1} \frac{d x}{1+x}\right|<\int_{0}^{1} x^{2 n} d x\) (d) Use part (c) to show that the sum of the given series is ln 2. Equation Transcription: < Text Transcription: 1/1.2+1/3.4+1/5.2+1/7.8+...=ln 2 1-x+x^2-x^3+...+x^2n-2-x^2n-1 1-1/2+1/3+1/4+...+1/2n-1-1/2n |1/1.2+1/3.4+1/5.6+...+1/(2n-1)(2n)-integral_0 ^1 dx/1+x|<integral_0 ^1 x^2n dx

Find all the solutions of the equation \(1+\frac{x}{2 !}+\frac{x^{2}}{4 !}+\frac{x^{3}}{6 !}+\frac{x^{4}}{8 !}+\cdots=0\) [Hint: Consider the cases \(x \geqslant 0\) and \(x<0\) separately.

Right-angled triangles are constructed as in the figure. Each triangle has height 1 and its base is the hypotenuse of the preceding triangle. Show that this sequence of triangles makes infinitely many turns around by showing that is a divergent series.

Consider the series whose terms are the reciprocals of the positive integers that can be written in base 10 notation without using the digit Show that this series is convergent and the sum is less than 90 .

(a) Show that the Maclaurin series of the function \(f(x)=\frac{x}{1-x-x^{2}} \text { is } \sum_{n=1}^{\infty} f_{x} x^{n}\) where \(f_{n}\) is the n th Fibonacci number, that is, \(f_{1}=1, f_{2}=1\), and \(f_{n}=f_{n-1}+f_{n-2}\) for \(n \geqslant 3\). Find the radius of convergence of the series. [Hint: Write \(x /\left(1-x-x^{2}\right)=c_{0}+c_{1} x+c_{2} x^{2}+\ldots\) and multiply both sides of this equation by \(1-x-x^{2}\).] (b) By writing \(f_{n}\) as a sum of partial fractions and thereby obtaining the Maclaurin series in a different way, find an explicit formula for the n th Fibonacci number. Equation Transcription: ? Text Transcription: f(x)=x/1-x-x^2 is sum_n=1 ^infinity f_x X^n f_n f_1 =1, f_2 = 1 f_n = f_n-1 + f_n-2 n geqslant 3 x/ (1-x-x^2 ) = c_0 + c_1 x + c_2 x^2 + … 1-x-x^2 f(x)

Let \(u=1+\frac{x^3}{3 !}+\frac{x^6}{6 !}+\frac{x^9}{9 !}+\cdots\) \(v=x+\frac{x^4}{4 !}+\frac{x^7}{7 !}+\frac{x^{10}}{10 !}+\cdots\) \(w=\frac{x^2}{2 !}+\frac{x^5}{5 !}+\frac{x^8}{8 !}+\cdots\) Show that \(u^3+v^3+w^3-3 u v w=1\).

Prove that if , the th partial sum of the harmonic series is not an integer. Hint: Let be the largest power of 2 that is less than or equal to and let be the product of all odd integers that are less than or equal to . Suppose that , an integer. Then The right side of this equation is even. Prove that the left side is odd by showing that each of its terms is an even integer, except for one.