Pranav said:Find the sum of the following series upto infinite terms:
$$\cot^{-1}\left(\frac{5}{\sqrt{3}}\right)+\cot^{-1}\left(\frac{9}{\sqrt{3}}\right)+\cot^{-1}\left(\frac{15}{\sqrt{3}}\right)+\cot^{-1}\left(\frac{23}{\sqrt{3}}\right)+\cdots$$
chisigma said:Since I'm more familiar with the function $\displaystyle \tan^{-1} x$ let me write the series as...
$\displaystyle S = \sum_{n=1}^{\infty} \tan^{-1} \frac{\sqrt{3}}{n^{2} + n + 3}\ (1)$
We can use the general formula...
$\displaystyle \sum_{n=1}^{\infty} \tan^{-1} \frac{c}{n^{2} + n + c^{2}} = \tan^{- 1} c\ (2)$
... obtaining...
$\displaystyle S = \tan^{-1} \sqrt{3} = \frac{\pi}{3}\ (3)$ The prove of (2) will be supplied in a successive post...
Kind regards
$\chi$ $\sigma$
Pranav said:Hi chisigma!
Thanks for participating, your answer is correct! I am interested in your proof for (2). :)
chisigma said:May be that the best for me is to open a math note dedicated to the series of inverse functions...
Kind regards
$\chi$ $\sigma$