Math 227 place out exam

Geometric series test

Converges if lrl<1. Converges to ar^k/(1-r)

Nth term test for divergence

If the limit does not equal zero, the series diverges.

If the limit equals zero, the series converges.

Integral test for convergence

Must be: positive, continuous, decreasing

P-series test

A series of the form $\sum \frac{1}{n^p}$ converges if p > 1 and diverges if $p \leq 1$.

Comparison test for convergence

(Draw the graph!)

Let 0<an<bn

If an diverges, bn diverges

If bn converges, an diverges

Limit comparison test

A method for determining the convergence or divergence of an infinite series. Given two series $\sum a_n$ and $\sum b_n$, if a_n > 0 for all n and the limit $\lim_{n \to \infty} \frac{a_n}{b_n} = c$ where 0 < c < \infty, then both series either converge or diverge together.

Alternating series test

A series of the form $\sum(-1)^{n}an$ converges if: 1) $a\_n$ is positive, 2) $a\_n$ is monotonically decreasing, and 3) $\lim\_{n \to \infty} a\_n = 0$. These criteria ensure convergence.

Ratio test for convergence

A method for determining the convergence or divergence of an infinite series. For a series $\sum a_n$, examine the limit $L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}$.

If L < 1, the series converges; if L > 1, the series diverges; if $L = 1$, the test is inconclusive.

e^x power series

cos(x) power series

sin(x) power series

1/(1-x)

1/(1+x)

arctan(x)

ln(1+x)