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Find the value of $\displaystyle \frac21- \frac{2^3}{3^2}+ \frac{2^5}{5^2}- \frac{2^7}{7^2}+ \cdots$ till infinite terms.

found this problem while integrating $\arctan\left(x\right)/x$ from $0$ to $2$ using its Taylor series expansion also tried writing $\arctan$ in complex $\log\left(\frac{1 + {\rm i}x}{1 - {\rm i}x}\right)$ form but could not get anywhere

is there any possible way to evaluate such harmonic geometric sequences

also can be seen that denominator is sum of odd squares which converges to $\pi^{2}/8$ if helpful

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    $\begingroup$ Why would you think that sum exists, that doesn't look like a series that converges... $\endgroup$
    – Macavity
    Commented Nov 10, 2022 at 18:06
  • $\begingroup$ diverges https://wolframalpha.com/input?i=sum+of+%282%5E%282n%2B1%29%2F%282n%2B1%29%5E2%29+*+%28-1%29%5En+from+0+to+infty $\endgroup$
    – Captain Chicky
    Commented Nov 10, 2022 at 18:14
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    $\begingroup$ A quick ratio test shows this series isn't convergent. The integral exists, but the integrand's Taylor series expansion does not have the radius of convergence needed to integrate across the whole interval $(0,2)$. $\endgroup$
    – Macavity
    Commented Nov 10, 2022 at 18:28
  • $\begingroup$ got it thanks! @Macavity $\endgroup$
    – Ashman Wadhawan
    Commented Nov 10, 2022 at 18:52
  • $\begingroup$ Related. $\endgroup$
    – J.G.
    Commented Nov 10, 2022 at 19:04

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