Use the ratio test to analyze ∑n=1∞n2n\sum_{n=1}^{\infty} \frac{n}{2^n}∑n=1∞2nn. Calculate limn→∞∣an+1an∣\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|limn→∞anan+1.
L=12L = \frac{1}{2}L=21; the series converges
L=1L = 1L=1; the test is inconclusive
L=2L = 2L=2; the series diverges
L=0L = 0L=0; the series converges