Series and Sequences

Geometric series equation

ar^(n-1) where r is a fraction

When does a geometric series converge and diverge

Converges when r is less than 1 and diverges when r is greater than or equal to 1 (THE rule)

What does a geometric sequence converge to

a/1-r

How do use divergence test

Find the limit as n goes to infinity. If the limit doesn’t equal 0, the series diverges. If it does, use another expression.

When to use divergence test

First try always. Easy to find so may as well

How to use integral test

Make series into function (n to x), then find integral of the series. Either both converge or both diverge.

What is the p series equation and what is the rule for convergence

1/n^P. Diverges if p<1 and converges if p>1

Explain comparison test

Compare to known value (p series, harmonic, geometric). If compared is smaller, must diverge. If larger, must converge.

Explain the Limit Comparison Test

Take the limit of the equation in question / a chosen equation. If the limit is above 0, both converge or both diverge. If the limit is infinity, comparison must diverge. If limit is 0, comparison must converge.

Explain the Ratio Test

Take the limit of a_n+1/a_n. Diverges if p>1 and converges absolutely if p<1.

Explain the Root Test

If both the numerator and the deminator are to the nth power, find the limit of the nth root of the function. Diverges if p>1 and converges if p<1.

What are the criteria for convergence of an alternating series

Find bn (where -1ⁿ=1). If bn decreases and the limit as n approaches infinity of bn= 0, the series converges. If the limit isn’t zero, the series diverges.

What is the equation for power series

C_nxⁿ

What are the convergence rules for power series

|x-a|< 1 converges to 1/1-x

What is the equation for the Taylor Series

fⁿ(a)/n! * (x-a)ⁿ

What are the steps to find the Taylor Series

Find fⁿ(a) (n being derivative number), and plug that into the equation to list out each term. Find consistencies to make a full Taylor Series