I'm struggling with the two following infinite series: $$ \sum_{i=1}^{\infty} i^2[(i+1)\Phi(i+1)-2i\Phi(i)+(i-1)\Phi(i-1)] $$ and $$ \sum_{i=1}^{\infty} i^2[\phi(i+1)-2\phi(i)+(i-1)\phi(i-1)] $$ where $\Phi(\cdot)$ and $\phi(\cdot)$ are the cdf and the pdf of a standard normal random variable. The former sum should be equal to $-5/12$, the latter $+1$, but I cannot prove it mathematically. Any help?
2026-03-29 18:34:34.1774809274
Sum of an infinite series involving the normal cdf and pdf
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