Home » Uncategorized » A trigonometric series

A trigonometric series

Evaluate the series

    \[\mathcal{S}= \sum_{n=1}^{\infty} \arcsin \left( \frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n}\sqrt{n+1}}\right)\]

Solution

The series telescopes since,

    \[\arcsin \left( \frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n}\sqrt{n+1}}\right) = \arcsin \sqrt{\frac{n}{n+1}} - \arcsin \sqrt{\frac{n-1}{n}}\]

Hence the limit equals \frac{\pi}{2}.

Read more

Leave a comment

Donate to Tolaso Network