Choose the Right Test
This test is best when you see factorials in the terms.
What is the Ratio Test?
∑ ar^n
This series converges if and only if |r| < 1.
What is a Geometric Series?
For alternating series, the error is less than or equal to this.
What is the first omitted term?
The series centered at 0.
What is a Maclaurin Series?
Use this test when your terms are rational functions and direct comparison is messy.
What is the Limit Comparison Test?
∑ 1/n^p
What is a p-series?
This converges when p > 1.
What is a p-series?
For the Integral Test, the remainder R_n is less than this.
What is ∫_n^∞ f(x) dx ?
Use this test to find the radius of convergence.
What is the Ratio/Root Test?
Best for series that look like 1/n^p or when you can easily prove an inequality.
What is the Direct Comparison Test?
∑ 1/n
What is the Harmonic Series?
This always diverges.
What is the Harmonic Series?
Taylor’s Inequality uses this symbol for the bound on the (n+1)th derivative.
What is M?
The interval where the power series converges.
What is the Interval of Convergence?
Use this when the function is positive, continuous, and decreasing (and has a nice antiderivative).
What is the Integral Test?
The series where most terms in the partial sum cancel.
What is a Telescoping Series?
If ∑ |an| converges, then the original series does this.
What is converges absolutely?
The remainder after n terms of the Taylor polynomial.
What is R_n(x)?
You get this by integrating or differentiating known series.
What is a Power Series Representation?
This test is specifically for alternating signs after you check absolute convergence.
What is the Alternating Series Test?
∑ (-1)n bn where bn > 0, decreasing, lim bn = 0
What is an Alternating Series?
If a series converges but its absolute series diverges, it is this.
What is conditionally convergent?
You must show this goes to 0 to prove a function equals its Taylor series.
What is the Remainder R_n(x)?
The error bound you use when the Taylor series alternates.
What is the Alternating Series Error Bound?