<s>[INST] Evaluate $1-\frac{1}{3\cdot 3}+\frac{1}{5\cdot 3^2}-\frac{1}{7\cdot 3^3}+\cdots$ My attempt
$$1-\frac{1}{3\cdot 3}+\frac{1}{5\cdot 3^2}-\frac{1}{7\cdot 3^3}+\cdots=\sum_{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)3^{n-1}}$$
By Leibniz alternative test for convergence. It is a convergent alternative series. How do I evaluate this limit?
 [/INST] $$
\sum_{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)3^{n-1}}=\sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)3^{n}}=\sqrt{3}\sum_{n=0}^\infty \frac{(-1)^{n}(1/\sqrt{3})^{2n+1}}{(2n+1)}
$$
$$
=\sqrt{3}\arctan(1/\sqrt{3}) = \frac{\sqrt{3}\pi}{6}
$$
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