Chapter 9

Question 1

Convergence of a Geometric Series

Answer 1

A geometric series with ratio “r” diverges if |r| ≥ 1. If 0 < |r| < 1, then the series converges to the sum ∞Σn=0 a(r)^n = a/(1 - r), 0 < |r| < 1

Question 2

nth Term Test for a Convergent Series

Answer 2

If ∞Σn=1 (an) converges, then lim(n -> ∞) an = 0

Question 3

nth Term Test for a Divergent Series

Answer 3

If lim(n -> ∞) an ≠ 0, then ∞Σn=1 (an) diverges

Question 4

The Integral Test

Answer 4

If “f” is positive, continuous, and decreasing for x ≥ 1 and an = f(n), then ∞Σn=1 (an) and 1∫∞ f(x)dx either both converge or both diverge

Question 5

P-series

Answer 5

1. converges if p > 1

2. diverges if 0 < p ≤ 1

Question 6

Direct Comparison Test

Answer 6

Let 0 < an ≤ bn for all “n”

1. If ∞Σn=1 (bn) converges, then ∞Σn=1 (an) converges.
2. If ∞Σn=1 (an) diverges, then ∞Σn=1 (bn) diverges.

Question 7

Limit Comparison Test

Answer 7

Suppose that an > 0, bn > 0, and lim(n -> ∞) an/bn = L where L is finite and positive. Then the two series Σan and Σbn either both converge or both diverge.

Question 8

Alternating Series Test

Answer 8

Let an > 0. The alternating series ∞Σn=1 [(-1)^n * (an)] and ∞Σn=1 [(-1)^n+1 * (an)] converge if the following two conditions are met:

1. lim(n -> ∞) an = 0
2. a(n+1) ≤ an, for all “n”

Question 9

Alternating Series Remainder

Answer 9

If a convergent series satisfies the condition a(n+1) ≤ an, then the absolute value of the remainder (Rn) involved in approximating the sum “S” by Sn is less than (or equal to) the first neglected term. That is, |S - Sn| = |Rn| ≤ a(n+1)

Question 10

Absolute Convergence

Answer 10

If the series Σ|an| converges, then the series Σan also converges.

Question 11

Definitions of Absolute and Conditional Convergence (2)

Answer 11

1. Σan is absolutely convergent if Σ|an| converges.

2. Σan is conditionally convergent if Σan converges, but Σ|an| diverges.

Question 12

nth Taylor Polynomial

Answer 12

Pn(x) = f(c) + f’(c)(x-c) + f’’(c)/2! (x-c)^2 + . . . + f^n(c)/n! (x-c)^n

Question 13

Taylor’s Theorem: Rn(x) = ?

Answer 13

f^n+1/[(n+1)!] * (x-c)^(n+1)

Question 14

Taylor’s Theorem: |Rn(x)| ≤ ?

Answer 14

|x-c|^[(n+1)]/[(n+1)!] * max|f^n+1|

Question 15

Ratio Test

Answer 15

Let Σan be a series with nonzero terms

1. Σan converges absolutely if lim(n -> ∞) |a(n+1)/an| < 1
2. lim(n -> ∞) an diverges if lim(n -> ∞) |a(n+1)/an| > 1 or lim(n -> ∞) |a(n+1)/an| = ∞
3. The Ratio Test is inconclusive if lim(n -> ∞) |a(n+1)/an| = 1

Question 16

Root Test

Answer 16

Let Σan be a series

1. Σan converges absolutely if lim(n -> ∞) n√|an| < 1
2. Σan diverges if lim(n -> ∞) n√|an| > 1 or lim(n -> ∞) n√|an| = ∞
3. The Ratio Test is inconclusive if lim(n -> ∞) n√|an| = 1

Question 17

Convergence of Power Series

1. when would the series converge only at c?
2. when would the series converge absolutely in an interval?
3. when would the series converge absolutely for all x

Answer 17

1. the limit is ∞
2. the limit is some x variable less than one (solve for x)
3. the limit is a number less than 1