Series Math Problem: Finding the Value of an Infinite Series

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The discussion focuses on evaluating the infinite series sum of the difference between the arctangent of consecutive integers, specifically \(\sum_{n=1}^{\infty} [\tan^{-1}(n+1) - \tan^{-1}(n)]\). The user expresses uncertainty about how to approach the problem and considers using the Taylor series expansion for \(\tan^{-1}(x)\) to assist in finding the solution. After some exploration, they realize that the series converges to \(\frac{\pi}{2}\), correcting their initial miscalculation of \(\frac{\pi}{4}\). The conversation highlights the importance of understanding series convergence and the properties of inverse trigonometric functions. Overall, the series converges to the expected value of \(\frac{\pi}{2}\).
DivGradCurl
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I'm just not so sure on how to approach this problem. Well, here it goes:

\sum _{n=1} ^{\infty} \left[ \tan ^{-1} (n+1) - \tan ^{-1} (n) \right] = \frac{\pi}{2}

I know that

\tan ^{-1} x = \sum _{n=0} ^{\infty} \left( -1 \right) ^n \frac{x^{2n+1}}{2n+1}

but I don't know if it can be useful to get to the answer above. I just need some tips. Any help is highly appreciated.
 
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Try writing out the first few terms.

P.S. I get \pi/4.
 
Oh... I see. By the way, you're right about the \pi/4.

Thanks
 
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