U10: Convergence & Divergence

Question 1

Geometric Series Test

Answer 1

Converge: |r| < 1
Diverge: |r| >= 1

FORMAT: Σar^n
CONVERGES AT: a /(1-r)

Question 2

nth Term Test

Answer 2

Converge: N/A
Diverge: lim(n→ ∞) a(n) ≠ 0

INCONCLUSIVE: a(n) = 0

Question 3

Integral Test

Answer 3

Converge: ∫a(n)dn converges
Diverge: ∫a(n)dn diverges

CONDITIONS:
1) Positive
2) Continuous
3) Decreasing

Question 4

p-Series Test

Answer 4

Converge: p > 1
Diverge: p <= 1

FORMAT: 1/(n^p)
HARMONIC SERIES: p=1 (diverges)

Question 5

Comparison Test

Answer 5

Converge: a larger series converges
Diverge: a smaller series diverges

Question 6

Limit Comparison Test

Answer 6

Converge: other series converges
Diverge: other series diverges

MATH: lim(n → ∞) (a(n) / b(n)) = L
CONDITIONS:
1) a(n) and b(n) are positive
2) L is finite and positive

Question 7

Alternating Series Test

Answer 7

Converge: see below
Diverge: fails both tests

ABSOLUTE: |a(n)| converges
CONDITIONAL:
1) lim(n → ∞) a(n) = 0
2) a(n+1) <= a(n)

FORMAT: Σ(-1)^n a(n)

Question 8

Ratio Test

Answer 8

Converge: < 1
Diverge: > 1 (including ∞)

MATH: | lim(n → ∞) (a(n+1)/a(n)) |
INCONCLUSIVE: = 1

Question 9

Alternating Series Error Bound

Answer 9

|S - S(n)| <= |a(n+1)|
(Absolute value of next term)

Question 10

Lagrange Error Bound

Answer 10

|Rn(x)| <=
( |x-c|^(n+1) / (n+1)! ) * max|f(n+1)(z)|

For Taylor Polynomials

Question 11

Taylor Polynomial terms

Answer 11

a(n) = ( f(n)(c) / n! ) * (x-c)^n

Question 12

Power Series Convergence Interval

Answer 12

1) Ratio test ( lim (n → ∞) |a(n+1)/a(n)| )
2) set absolute value less than 1
3) algebraically find radius
4) add/subtract from center for interval

Converge: inside radius (absolute)
Diverge: outside radius
INCONCLUSIVE: at radius (test each)

Question 13

e^x Taylor Approximation

Answer 13

Σ x^n / n!

n=0, ∞

Question 14

sin(x) Taylor Approximation

Answer 14

Σ(-1)^n ( x^(2n+1) / (2n+1)! )

n=0, ∞

Question 15

cos(x) Taylor Approximation

Answer 15

Σ(-1)^n ( x^(2n) / (2n)! )

n=0, ∞