If S is the sum of the first 10 terms of the series ^-113+^-117+^-1113+^-1121+, then (S) i…

Question

If $S$ is the sum of the first 10 terms of the series $	an^{-1}\dfrac13+	an^{-1}\dfrac17+	an^{-1}\dfrac{1}{13}+	an^{-1}\dfrac{1}{21}+\cdots$, then $	an(S)$ is equal to:

Accepted Answer

The general term is $\tan^{-1}\left(\dfrac{1}{1+n(n+1)}\right)=\tan^{-1}(n+1)-\tan^{-1}n$ for $n=1,2,\ldots$. Summing the first 10 terms telescopes:

$$S=\tan^{-1}(11)-\tan^{-1}(1)=\tan^{-1}\left(\frac{11-1}{1+11}\right)=\tan^{-1}\frac{5}{6}.$$

$$\tan(S)=\frac56.$$

Solutions · 1