There is a known expansion for the Dirac delta function in terms of the Legendre polynomials as
$\delta(x) = \sum_{k = 0}^{\infty} (-1)^k \frac{(4k + 1) (2k)!}{2^{2k + 1} (k!)^2} P_{2k}(x)$.
I would like to verify this identity numerically. We know that $\delta(1) = 0$, so inserting $x = 0$ in the right-hand side of the above must result in zero. If we add the first two hundreds of the terms in the sum using Mathematica, results in $7.994$. We expect as we add more terms, the value of the sum decreases and approaches zero, however, for the first three hundreds terms, we obtain $9.784$; and it becomes larger as we add more terms. How will this series approach zero for an infinite number of terms?