Determine whether the series is convergent or divergent.∑from k=1to ∞ (ke^-9k)

Answer 1

since e^(-9k) goes to zero much faster than k grows, the series converges.
Use the ratio test.
A_(n+1)/A_n = (k+1)e^(-9k-9)/ke^(-9k)
= (k+1)/k * e^-9