Chapter 10

Question 1

Nth term

Answer 1

Lim as n → ∞ of aₙ = 0, inconclusive

Lim as n → ∞ of aₙ ≠ 0, diverges

Question 2

Geometric

Answer 2

needs to be geometric
|r|<1, converges
|r| ≥1, diverges
S = a/1-r

Question 3

P-series

Answer 3

p > 1, converges
0 < p < 1, diverges
harmonic series 1/n diverges

Question 4

Integral

Answer 4

f(n) must be continuous, positive, and decreasing
if integral converges, converges
if integral diverges, diverges

Question 5

Direct Comparison

Answer 5

0 ≤ aₙ ≤ bₙ
if bₙ converges, aₙ converges
if aₙ diverges, bₙ diverges

Question 6

Limit Comparison

Answer 6

aₙ > 0 and bₙ > 0
lim n—> ∞ aₙ/bₙ = L
L has to be finite and positive
If above is true, both series converge or diverge

Question 7

Ratio

Answer 7

lim n —> ∞ |aₙ₊₁/aₙ| < 1, converge

lim n —> ∞ |aₙ₊₁/aₙ| > 1, diverge

lim n —> ∞ |aₙ₊₁/aₙ| = 1, inconclusive

Question 8

Alternating Series

Answer 8

If |aₙ₊₁| < |aₙ| and lim n —> ∞ aₙ = 0, converges

Question 9

E^x polynomial

Answer 9

1+x+x^2/2!+x^3/3!+x^4/4! … + x^n/n!

Question 10

1/1-x polynomial

Answer 10

1+x+x^2+x^3 + … + x^n

Question 11

sin(x) polynomial

Answer 11

x-x^3/3!+x^5/5!-x^7/7!+ … + (-1)^n • x^2n+1/(2n+1)!

Question 12

cos(x) polynomial

Answer 12

1-x^2/2!+x^4/4!-x^6/6!+ … + (-1)^n • x^2n/(2n)!