# 9.4 | Tests for Convergence

The Alternating Series Test: Given a series of the form
$$\sum (-1)^n a_n$$
with $$a_n >0$$, if

• $$\displaystyle \lim_{n \rightarrow \infty} a_n = 0$$
• $$a_n$$ is decreasing. $$(a_{n+1} \leq a_n)$$

then the series converges.

The (Direct) Comparison Test: Given a series $$\sum a_n$$ with $$a_n >0$$ choose a comparison series $$\sum b_n$$ with $$b_n >0$$.

• If $$a_n \leq b_n$$ and $$\sum b_n$$ converges, then so does $$\sum a_n$$.
• If $$b_n \leq a_n$$ and $$\sum b_n$$ diverges, then so does $$\sum a_n$$.
The Limit Comparison Test: Given a series $$\sum a_n$$ with $$a_n >0$$ choose a comparison series $$\sum b_n$$ with $$b_n >0$$.
If
$$\lim_{n \rightarrow \infty} \frac{a_n}{b_n}$$
is a positive number then either both series converge or both diverge. Otherwise, the test is inconclusive.
A series $$\sum a_n$$ is absolutely convergent provided $$\sum |a_n|$$ converges. That is, a series converges absolutely if the same series with absolute values around the terms of the sum converges.
If a series converges absolutely, then it converges.
Ratio Test: Given $$\sum a_n$$ with $$a_n \not = 0$$ then

• If $$\displaystyle \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n}\right|<1$$, the series converges absolutely.
• If $$\displaystyle \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n}\right|>1$$, the series diverges.
• If $$\displaystyle \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n}\right|=1$$, the test is inconclusive.
Root Test: Given $$\sum a_n$$ then

• If $$\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{\left| a_n\right|}<1$$, the series converges absolutely.
• If $$\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{\left| a_n\right|}>1$$, the series diverges.
• If $$\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{\left| a_n\right|}=1$$, the test is inconclusive.