Convergence Tests

Question 1

Nth Term Test

Answer 1

If limit ≠ 0, the series diverges
If limit = 0, further investigation is needed
(CANNOT determine convergence)

Question 2

Integral Test

Answer 2

aₙ = f(n)
Function must be continuous, positive, decreasing, and readily integrable

(i) Converges if integral converges
(ii) Diverges if integral diverges

Question 3

Basic Comparison Test (BCT)

Answer 3

The comparison series is often a geometric series or p-series

(i) If Σbₙ converges and aₙ ≤ bₙ, then aₙ converges
(ii) If Σbₙ diverges and aₙ ≥ bₙ, then aₙ diverges

Question 4

Limit Comparison Test (LCT)

Answer 4

Use higher power of rational function and compare to a p-series that creates the same power in both numerator and denominator.

If lim aₙ/bₙ = C > 0, then either both converge or both diverge.

Question 5

Ratio Test

Answer 5

Inconclusive if L = 1
Useful if an involves factorials and/or nth powers

If lim aₙ₊₁/aₙ= L,
(i) Converges if L < 1
(ii) Diverges if L > 1 or = ထ

Question 6

Root Test

Answer 6

Inconclusive if L = 1
Useful if aₙ involves nth powers

If lim n√|aₙ| = L,
(i) Converges if L < 1
(ii) Diverges if L > 1 or = ထ

Question 7

Geometric Series

Answer 7

Σar^n-1

(i) Converges to (S = a₁/1-r) if |r| < 1
(ii) Diverges if |r| ≥ 1

Question 8

P-series

Answer 8

Σ1/n^p

(i) Converges if p > 1
(ii) Diverges if p ≤ 1

Question 9

Alternating Series

Answer 9

Σ(-1)^n aₙ,
aₙ > 0

Converges if
(i) aₖ ≥ aₖ₊₁ for every k (can usually assume this is true)
(ii) limit aₙ = 0

Question 10

Σ|aₙ|
(absolute value)

Answer 10

Useful for series that contain both positive and negative terms

If Σ|aₙ| converges, Σ aₙ converges

Question 11

McLaurin Series

Answer 11

Create a chart with
(i) list of n values
(ii) first derivative, f’(x)
(iii) evaluate first derivative at x = 0, f’(0)
(iv) multiply by x^n/n!
(v) look at sum, find pattern, generate series

Question 12

Taylor Series

Answer 12

Create a chart with
(i) list of n values
(ii) first derivative, f’(x)
(iii) evaluate first derivative at x = c, f’(c)
(iv) multiply by (x-c)^n/n!
(v) look at sum, find pattern, generate series

Question 13

LeGrange Error

Answer 13

Create chart with
(i) list of n values up to given degree + remainder
(ii) first derivative, f’(x)
(iii) evaluate first derivative at x = c, f’(c)
(iv) multiply by (x-c)^n/n!
(v) generate Taylor polynomial + remainder

(i) Plug given number and approximated number into remainder
(i) Largest one is the error