Calc 2 Midterm 3 Limit Tests

Sequence

infinite group of numbers

lim k →∞ =L → L is point of convergence

looking for convergence or divergence

Series

sum of the terms of a sequence

lim k → ∞ aₖ

looking for convergence or divergence

use tests to determine

Geometric Series

|r|≥1 → series diverges

|r|<1 → series converges

if sereis converges then a/(1-r)= S, the sum of the series

P-Series Test:

P≤1 → series diverges

P>1 → series converges

harmonic series diverges

Divergence Test

lim k → ∞ aₖ ≠ 0 then series diverges

if lim k → ∞ aₖ = 0 then test is inconclusive

Integral Test

for this series,

f(x)=aₖ=aₓ

if the limit of f(x) converges, than the series of aₖ also converges

if the limit of f(x) diverges, than the series of aₖ also diverges

→ use this test if it is easy to integrate

Must meet these terms:

• is f(x) continuos on x≥1

• is f(x) positive on x≥1?

• is f(x) decreasing on x≥1

Limit Comparison Test

for ∑ aₙ find a comparable series, bₙ

needs same highest power of k

find lim n → ∞ aₙ / bₙ = L

if L is a positive finite number, then aₙ and bₙ both converge or diverge

→ think p-series for bₙ because easy to tell if converge or diverge

Ratio Test

aₖ≠0

→ use if factorial or k in exponent

if |r|<1 then ∑ aₖ converges

if |r|>1 then ∑ aₖ diverges

if |r|=1 then test is inconclusive

Alternating Series Test

1. need alternating series (-1)^k etc

2. lim k→∞ aₖ=0 → not lim of entire ∑, just aₖ

3. {aₖ} is decreasing → (aₖ₊₁ / aₖ)≤1 → that is the r value

If the series passes these 3 tests than it converges

• if it passes 1 and 3, but not 2, then it diverges

• if it passes 1 and 2, but not 3, then it is inconclusive