Geometric Series
Key Questions

You can find the common ratio
#r# by finding the ratio between any two consecutive terms.#r=a_1/a_0=a_2/a_1=a_3/a_2=cdots=a_{n+1}/a_n=cdots# For the geometric series
#sum_{n=0}^infty(1)^n{2^{2n}}/5^n# , the common ratio#r=a_1/a_0={2^2/5}/{1}=4/5# 
The sum of a convergent geometric series
#sum_{n=0}^{infty}ar^n# is#frac{a}{1r}# . Since#a=8# and#r=1/2# in the posted geometric series, the sum is#frac{8}{1frac{1}{2}}=16# . 
It's the sum of a sequence of terms where each term equals the previous one multiplied by a given ratio (e.g.
#1 + 2 + 4 + 8...# ).
Questions
Tests of Convergence / Divergence

Geometric Series

Nth Term Test for Divergence of an Infinite Series

Direct Comparison Test for Convergence of an Infinite Series

Ratio Test for Convergence of an Infinite Series

Integral Test for Convergence of an Infinite Series

Limit Comparison Test for Convergence of an Infinite Series

Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series

Infinite Sequences

Root Test for for Convergence of an Infinite Series

Infinite Series

Strategies to Test an Infinite Series for Convergence

Harmonic Series

Indeterminate Forms and de L'hospital's Rule

Partial Sums of Infinite Series