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How do I find the sum $$\sum_{n=0}^{\infty}(-1)^n P_n(x)$$ where $P_n$ are the $n$th order Legendre polynomials? I tried using the generating function but I was not able to arrive at an answer. Any hints appreciated.

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  • $\begingroup$ You only need to evaluate the generating function at $t=-1$. $\endgroup$ – conditionalMethod Nov 8 at 14:41
  • $\begingroup$ @conditionalMethod Its not allowed to substitute the value of $t$ in the generating function $\endgroup$ – physics123 Nov 8 at 14:51
  • $\begingroup$ I don't know who will stop me. Oops, I just did it. Wait, I will even dare to write $\sum_{n=0}^{\infty}(-1)^nP_n(x)=\frac{1}{\sqrt{2+2x}}$. Send the cops. $\endgroup$ – conditionalMethod Nov 8 at 14:56
  • $\begingroup$ @conditionalMethod The generating function is derived by assuming $t$ is very small. The cops are on their way! $\endgroup$ – physics123 Nov 8 at 15:06

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