Sequences

Sequence

A function whose domain is the set of positve integers

{a_n}

Absolute Value theorem

Monotonic

A sequence that is nondecreasing OR nonincreasing

Bounded Monotonic sequences

If sequence {a_n} is bounded and monotonic, then it converges

Infinite series

Geometric series

Convergence of a geometric series

When ratio |r|>=1 it diverges. When |r|<1 the series converges to the sum (a/(1-r)), or (first term)/(1-ratio)

Telescopic Series

1. do partial fractions to split it up

2. List the terms ending at n

3. Cancel

4. Take the limit to infinity and find if it is convergent

nth term test

Integral test

Let f(x) = ___

f must be positive, continuous, and decreasing

solve integral from infinity to 1, if the integral diverges, series diverges.

P-series

converges for p>1 diverges for 0<p<=1

Direct comparison test

if 0<a_n<=b_n for all n

• if b_n converges, then a_n converges

• if b_n diverges, then a_n diverges

Limit comparison test

if a_n>0 and b_n>0 and lim_n→infinity(a_n/b_n)=L

then, either both converge or diverge

Alternating Series

Alternating Series Test

if nth term test=0 and it is monotonic