For factorial sums, the key identity is kk!=(k+1)!-k!, which makes sum(kk!, k=1 to n)=(n+1)!-1. Related: $\frac{k}{k+1}$!=1/k!-$\frac{1}{k+1}$!. For sums involving 1/k!: sum(1/k!, k=0 to inf)=e, but finite sums don't simplify to a closed form. For sums involving binomial coefficients: use derivatives or integrals of (1+x)^n. sum(kC(n,k))=n2^(n-1) (differentiate (1+x)^n at x=1). Arithmetic-Geometric series use the S-rS trick: multiply by common ratio, subtract, and simplify the resulting GP.