Alternating series test Help

Aug 2016
46
0
illinois
if you have an alternating series with (-1)^(n+1)/2n+1, why do you ignore the (-1)^(n+1) when you check if the limit is 0 and if it is decreasing and only look at 1/(2n+1).
 
Oct 2016
77
35
Melbourne
An alternating series is DEFINED as something like $\displaystyle \begin{align*} \sum{ \left( -1 \right) ^n\, a_n } \end{align*}$ where $\displaystyle \begin{align*} a_n geq 0 \end{align*}$ for all n. This is to ensure that the series actually does alternate.

As the series alternates, each term will either add or subtract something to the total amount. To ensure that the series converges, each term can not have any "extra" effect to counteract the previous terms that have been added, which means that its absolute value has to be smaller than the previous ones. Notice that $\displaystyle \begin{align*} \left| \left( -1 \right) ^n\,a_n \right| = a_n \end{align*}$ as $\displaystyle \begin{align*} a_n \geq 0 \end{align*}$, so really only the $\displaystyle \begin{align*} a_n \end{align*}$ terms are needed to be decreasing to ensure convergence.