Given that $-\frac{\pi}{2} < x < \frac{\pi}{2}$, for what values of $x$ does the series
$$1 – \tan^2 x + \tan^4 x – \cdots$$
have a limiting sum? Hence find the limiting sum.
I don't have any real working because I don't know where to start.
I assume you put it into terms of $\cos x$ and use $\frac{a}{1-r}$ to get limit, but I don't know how to get there.
Do I assume a value for x and guess and check?
I'm new and learning how to use Stack Exchange but any help is appreciated with the question and/or website.
Best Answer
The given series can be rewritten as \begin{align*} \sum_{k=0}^{\infty}(-1)^{k}\tan^{2k}(x) = 1 - \tan^{2}(x) + \tan^{4}(x) - \ldots \end{align*}
which is a geometric series whose common ratio is given by $-\tan^{2}(x)$. Thus it converges when \begin{align*} |-\tan^{2}(x)| < 1 \Longleftrightarrow |\tan(x)| < 1 \Longleftrightarrow -\frac{\pi}{4} < x < \frac{\pi}{4} \end{align*}