Evaluate the series ∑n=1∞n(13)n\sum_{n=1}^{\infty} n \left(\frac{1}{3}\right)^n∑n=1∞n(31)n using summation by parts (Abel's summation formula).
34\frac{3}{4}43
916\frac{9}{16}169
316\frac{3}{16}163
Diverges