Alternating Series of Real Numbers

Definition: Let $(a_n)_{n=1}^{\infty}$ be a strictly positive sequence of real numbers ($a_n > 0$ for all $n \in \mathbb{N}$). Then the corresponding alternating series for this sequence is a series of the form $\displaystyle{\sum_{n=1}^{\infty} (-1)^{n+1} a_n}$.

Alternatively one can define an alternating series such that for the sequence of terms $(a_n)_{n=1}^{\infty}$ we have that $a_na_{n+1} < 0$ for all $n \in \mathbb{N}$ which implies that every alternating term in the sum is negative while the remaining terms are positive.

Suppose that we have an alternating sequence:

Then the corresponding alternating series can be thought of as summing all of the positive terms and then adding the sum of all of the negative terms:

For example, the following series are alternating series:

(1)

\begin{align} \quad \sum_{n=1}^{\infty} (-1)^{n+1} \left ( \frac{1}{n} \right ) = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} + ... \end{align}

(2)

\begin{align} \quad \sum_{n=1}^{\infty} (-1)^{n+1} \left ( \frac{1}{n^2} \right ) = 1 - \frac{1}{4} + \frac{1}{9} - \frac{1}{16} + \frac{1}{25} + ... \end{align}

Note that $(-1)^n = \cos ((n+1)\pi)$ for all $n \in \mathbb{N}$ so the two alternating series could be written as $\displaystyle{\sum_{n=1}^{\infty} \frac{\cos ((n+1)\pi)}{n}}$ and $\displaystyle{\sum_{n=1}^{\infty} \frac{\cos ((n+1)\pi)}{n^2}}$. In fact, there are many identities for $(-1)^n$ so the reader should always be aware of such identities that may pop up in numerical series.