If ^-1()=^-12+^-18+^-118+^-132+ up to 100 terms, then is

Question

If $\cot^{-1}(\alpha)=\cot^{-1}2+\cot^{-1}8+\cot^{-1}18+\cot^{-1}32+\cdots$ up to 100 terms, then $\alpha$ is

Accepted Answer

The general term $T_n=\cot^{-1}(2n^2)=\tan^{-1}\left(\dfrac{1}{2n^2}\right)=\tan^{-1}\left(\dfrac{(2n+1)-(2n-1)}{1+(2n+1)(2n-1)}\right)=\tan^{-1}(2n+1)-\tan^{-1}(2n-1)$.

Telescoping over $n=1$ to $100$:

$$\sum T_n=\tan^{-1}(201)-\tan^{-1}(1)=\tan^{-1}\left(\frac{201-1}{1+201}\right)=\tan^{-1}\left(\frac{200}{202}\right).$$

$$\cot^{-1}(\alpha)=\cot^{-1}\left(\frac{202}{200}\right)\ \Rightarrow\ \alpha=\frac{202}{200}=\frac{101}{100}=1.01.$$

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