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.