Alternating Series Test Alternating Series Test
Introduction
An alternating series is defined as:
\sum_{n=0}^{\infty} (-1)^n a_n \quad \text{or} \quad \sum_{n=1}^{\infty} (-1)^{n+1} a_n
where
a_n
are positive terms. Thus, each term in the series alternates sign.
For an alternating series to converge, it must satisfy the following conditions:
The sequence, a_n
, must be decreasing for all n
sufficiently large.
The sequence must approach zero:
\lim_{n \to \infty} a_n = 0
Estimating Sums and Error Bounds
For converging, alternating series, the absolute value of the remainder after
n
terms is less than or equal to the first neglected term.
R_n
: The remainder after
n
terms.
\begin{align*}
R_n &= |S - S_n| \\
&= |\sum_{k=1}^{\infty} (-1)^{k} a_k - \sum_{k=1}^{n} (-1)^{k} a_k| \\
&= |\sum_{k=n+1}^{\infty} (-1)^{k} a_k| \leq a_{n+1}
\end{align*}
where
S
is the actual sum and
S_n
is the sum of the first
n
terms.
Conditional vs Absolute Convergence
Conditional Convergence:
A series is conditionally convergent if it converges, but the series created by summing the absolute value of its terms diverges:
\sum_{n=1}^{\infty} a_n \text{ converges} \quad \sum_{n=1}^{\infty} |a_n| \text{ diverges}
Absolute Convergence:
A series is absolutely convergent if the series created by summing the absolute value of its terms converges:
\sum_{n=1}^{\infty} |a_n| \text{ converges}
Examples
Example 1: Alternating Harmonic Series
\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}
This series converges by the Alternating Series Test since:
a_n = \frac{1}{n}
is positive and decreasing.
\lim_{n \to \infty} a_n = 0
. However, it is conditionally convergent, since:
\sum_{n=1}^{\infty} \frac{1}{n} \text{ diverges}
Example 2: Divergent Series \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}
This series absolutely converges since:
\sum_{n=1}^{\infty} \frac{1}{n^2} \text{ converges}
by the p-series test (with p = 2).