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.

E 9.4 Exercises