 9.6.1: In Exercises 14, verify the formula.\n 1! n 2! n 1nn 1
 9.6.2: In Exercises 14, verify the formula.2k 2! 2k! 1 2k2k 1
 9.6.3: In Exercises 14, verify the formula.1 3 5 . . . 2k 1 2k! 2k k!
 9.6.4: In Exercises 14, verify the formula.
 9.6.5: In Exercises 510, match the series with the graph of its sequence o...
 9.6.6: In Exercises 510, match the series with the graph of its sequence o...
 9.6.7: In Exercises 510, match the series with the graph of its sequence o...
 9.6.8: In Exercises 510, match the series with the graph of its sequence o...
 9.6.9: In Exercises 510, match the series with the graph of its sequence o...
 9.6.10: In Exercises 510, match the series with the graph of its sequence o...
 9.6.11: In Exercises 11 and 12, (a) verify that the series converges. (b) U...
 9.6.12: In Exercises 11 and 12, (a) verify that the series converges. (b) U...
 9.6.13: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.14: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.15: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.16: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.17: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.18: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.19: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.20: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.21: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.22: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.23: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.24: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.25: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.26: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.27: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.28: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.29: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.30: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.31: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.32: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.33: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.34: In Exercises 1334, use the Ratio Test to determine the convergence ...
 9.6.35: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.36: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.37: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.38: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.39: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.40: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.41: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.42: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.43: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.44: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.45: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.46: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.47: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.48: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.49: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.50: In Exercises 3550, use the Root Test to determine the convergence o...
 9.6.51: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.52: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.53: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.54: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.55: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.56: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.57: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.58: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.59: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.60: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.61: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.62: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.63: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.64: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.65: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.66: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.67: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.68: In Exercises 5168, determine the convergence or divergence of the s...
 9.6.69: In Exercises 6972, identify the two series that are the same
 9.6.70: In Exercises 6972, identify the two series that are the same
 9.6.71: In Exercises 6972, identify the two series that are the same
 9.6.72: In Exercises 6972, identify the two series that are the same
 9.6.73: In Exercises 73 and 74, write an equivalent series with the index o...
 9.6.74: In Exercises 73 and 74, write an equivalent series with the index o...
 9.6.75: In Exercises 75 and 76, (a) determine the number of terms required ...
 9.6.76: In Exercises 75 and 76, (a) determine the number of terms required ...
 9.6.77: a1 1 2 , an1 4n 1 3n 2 an
 9.6.78: a1 2, an1 2n 1 5n 4 an
 9.6.79: a1 1, an1 sin n 1 n an
 9.6.80: a1 1 5 , an1 cos n 1 n an
 9.6.81: a1 1 3 , an1 1 1 n an
 9.6.82: a1 1 4 , an1 n an
 9.6.83: In Exercises 8386, use the Ratio Test or the Root Test to determine...
 9.6.84: In Exercises 8386, use the Ratio Test or the Root Test to determine...
 9.6.85: In Exercises 8386, use the Ratio Test or the Root Test to determine...
 9.6.86: In Exercises 8386, use the Ratio Test or the Root Test to determine...
 9.6.87: In Exercises 8792, find the values of for which the series converge...
 9.6.88: In Exercises 8792, find the values of for which the series converge...
 9.6.89: In Exercises 8792, find the values of for which the series converge...
 9.6.90: In Exercises 8792, find the values of for which the series converge...
 9.6.91: In Exercises 8792, find the values of for which the series converge...
 9.6.92: In Exercises 8792, find the values of for which the series converge...
 9.6.93: State the Ratio Test.
 9.6.94: State the Root Test
 9.6.95: You are told that the terms of a positive series appear to approach...
 9.6.96: The graph shows the first 10 terms of the sequence of partial sums ...
 9.6.97: Using the Ratio Test, it is determined that an alternating series c...
 9.6.98: Using the Ratio Test, it is determined that an alternating series c...
 9.6.99: Prove Property 2 of Theorem 9.17.
 9.6.100: Prove Theorem 9.18. (Hint for Property 1: If the limit equals choos...
 9.6.101: In Exercises 101104, verify that the Ratio Test is inconclusive for...
 9.6.102: In Exercises 101104, verify that the Ratio Test is inconclusive for...
 9.6.103: In Exercises 101104, verify that the Ratio Test is inconclusive for...
 9.6.104: In Exercises 101104, verify that the Ratio Test is inconclusive for...
 9.6.105: Show that the Root Test is inconclusive for the pseries n1 1 np.
 9.6.106: Show that the Ratio Test and the Root Test are both inconclusive fo...
 9.6.107: Determine the convergence or divergence of the series when (a) (b) ...
 9.6.108: Show that if is absolutely convergent, then n1 an n1 an.
 9.6.109: Read the article A Differentiation Test for Absolute Convergence by...
 9.6.110: Is the following series convergent or divergent?1 1 2 19 7 2! 32 19...
 9.6.111: Show that if the series converges, then the series a1 a2 2 a3 3 . ....
Solutions for Chapter 9.6: The Ratio and Root Tests
Full solutions for Calculus  9th Edition
ISBN: 9780547167022
Solutions for Chapter 9.6: The Ratio and Root Tests
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.6: The Ratio and Root Tests includes 111 full stepbystep solutions. Calculus was written by and is associated to the ISBN: 9780547167022. Since 111 problems in chapter 9.6: The Ratio and Root Tests have been answered, more than 61103 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Calculus , edition: 9.

Absolute minimum
A value ƒ(c) is an absolute minimum value of ƒ if ƒ(c) ? ƒ(x)for all x in the domain of ƒ.

Binomial theorem
A theorem that gives an expansion formula for (a + b)n

Bounded above
A function is bounded above if there is a number B such that ƒ(x) ? B for all x in the domain of ƒ.

Cofunction identity
An identity that relates the sine, secant, or tangent to the cosine, cosecant, or cotangent, respectively

Identity matrix
A square matrix with 1’s in the main diagonal and 0’s elsewhere, p. 534.

Identity properties
a + 0 = a, a ? 1 = a

Infinite sequence
A function whose domain is the set of all natural numbers.

Jump discontinuity at x a
limx:a  ƒ1x2 and limx:a + ƒ1x2 exist but are not equal

Linear combination of vectors u and v
An expression au + bv , where a and b are real numbers

Linear factorization theorem
A polynomial ƒ(x) of degree n > 0 has the factorization ƒ(x) = a(x1  z1) 1x  i z 22 Á 1x  z n where the z1 are the zeros of ƒ

Linear regression equation
Equation of a linear regression line

Multiplication principle of probability
If A and B are independent events, then P(A and B) = P(A) # P(B). If Adepends on B, then P(A and B) = P(AB) # P(B)

Natural logarithm
A logarithm with base e.

Parametric equations
Equations of the form x = ƒ(t) and y = g(t) for all t in an interval I. The variable t is the parameter and I is the parameter interval.

Present value of an annuity T
he net amount of your money put into an annuity.

Quadratic formula
The formula x = b 2b2  4ac2a used to solve ax 2 + bx + c = 0.

Quartic function
A degree 4 polynomial function.

Reflection through the origin
x, y and (x,y) are reflections of each other through the origin.

Variation
See Power function.

Vertex of a parabola
The point of intersection of a parabola and its line of symmetry.