Series Convergence Tests

A series which can be written in the form
$$\sum_{n = 1}^{\infty} \frac{1}{n}$$
is called a harmonic series and diverges.
A geometric series is a series of the form
$$\sum_{n = 1}^{\infty} ar^{n-1} = a + ar + ar^2 + \cdots$$
and
$$\sum_{n =1}^{\infty} ar^{n-1} = \begin{cases} \frac{a}{1-r}, & \; |r| <1 \\ \\ \text{divergent}, & \; \text{otherwise} \end{cases}$$
A series which can be written in the form
$$\sum_{n=1}^{\infty} \frac{1}{n^p}$$
is called a $$p$$-series and
$$\sum_{n=1}^{\infty} \frac{1}{n^p} = \begin{cases} \text{convergent}, & \; \text{ if } p > 1 \\ \text{divergent}, & \; \text{ if } p \leq 1 \end{cases}$$
The Divergence Test: If $$\displaystyle \lim_{n \rightarrow \infty} a_n$$ does not exist or $$\displaystyle \lim_{n \rightarrow \infty} a_n \not = 0$$, then the series $$\displaystyle \sum_{n = 1}^{\infty} a_n$$ is divergent.
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 Integral Test: Let $$f(x)$$ be a positive, continuous, decreasing function on $$[1, \infty)$$ with $$f(n) = a_n$$ for $$n \geq 1$$. Then
$$\sum_{n = 1}^{\infty} a_n \text{ and } \int_{1}^{\infty} f(x) \; dx$$
either both converge or both diverge.
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.