Let's determine the sum of the following infinite series:

$$
1-\frac{1}{a}+\frac{1}{a^{2}}-\frac{1}{a^{3}}+\frac{1}{a^{4}}-\frac{1}{a^{5}}+\ldots
$$

if $a$ is a positive number greater than one. What will the sum be as the value of $a$ approaches 1?
The given progression can also be written as:

$$
\begin{gathered}
\left(1+\frac{1}{a^{2}}+\frac{1}{a^{4}}+\ldots\right)-\left(\frac{1}{a}+\frac{1}{a^{3}}+\frac{1}{a^{5}}+\ldots\right)= \\
\frac{a^{2}}{a^{2}-1}-\frac{a}{a^{2}-1}=\frac{a}{a+1}
\end{gathered}
$$

to \(\frac{1}{2}\).

If the value of \(a\) approaches 1, then the sum of the progression approaches the value of

\(\frac{a}{a+1}\) which is \(\frac{1}{2}\).