# Summation of Legendre polynomial series

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.

• You only need to evaluate the generating function at $t=-1$. – conditionalMethod Nov 8 at 14:41
• @conditionalMethod Its not allowed to substitute the value of $t$ in the generating function – physics123 Nov 8 at 14:51
• 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. – conditionalMethod Nov 8 at 14:56
• @conditionalMethod The generating function is derived by assuming $t$ is very small. The cops are on their way! – physics123 Nov 8 at 15:06