The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { \alpha } }$ converges if and only if A) $\alpha 1$ E) $\alpha \geq 1$ F) $\alpha - 1$

Question

The series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { \alpha } }$ converges if and only if A) $\alpha < 1$ B) $- 1 < \alpha < 1$ C) $\alpha \leq 1$ D) $\alpha > 1$ h. $- 1 > \alpha > 1$ E) $\alpha \geq 1$ F) $\alpha < - 1$ G) $\alpha > - 1$

Quizplus · Accepted Answer

The series converges if $\alpha > 1$, according to the p-series test, which states that the series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \leq 1$.

The Series ∑n=1∞1nα\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { \alpha } }∑n=1∞​nα1​

The Series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { \alpha } }$