Leibniz convergence

Question

I must demostrate the convergence of $pi$ in Leibniz series: \begin{equation} \sum_{i=1}^{+\infty}\frac{(-1)^i}{2i+1} \end{equation} I need to resolve it with this method $\displaystyle\lim_{x\to\infty}\frac{a_{n+1}{a_n}$.

I kown the solution, it is $\displaystyle\frac{\pi}{4}$, but i dont have the way.

Thanks

Answer 1

HINT

Recall that by Leibniz test for convergence of alternating series we need to check that

• $a_i\to 0$
• $|a_i|$ decreases monotonically