Question

∑（k=n,∞)(1-p)^(k-1)=(1-p)^(n-1)∑(k=0,∞)(1-p)^k

Answer 1

m=k-n,k = n+m,k=n->∞,m=0->∞.

∑（k=n,∞)(1-p)^(k-1)=∑（m=0,∞)(1-p)^(n+m-1)

=(1-p)^n∑（m=0,∞)(1-p)^(m-1)

=(1-p)^n∑（k=0,∞)(1-p)^(k-1)