Which of the three series below converges?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$
2) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 1.1 } }$
3) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 09 } }$
A)1, 2
B)1
C)None
D)3
E)2
F)2, 3
G)1, 2, 3
H)1, 3

Question

Which of the three series below converges?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 1.1 } }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 09 } }$ 
A)1, 2
B)1
C)None
D)3
E)2
F)2, 3
G)1, 2, 3
H)1, 3

Quizplus · Accepted Answer

The series in option 2 converges due to the p-test for convergence, which states that a series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$. Here, $p = 1.1 > 1$, so the series converges. The series in options 1 and 3 diverge because they do not satisfy the condition $p > 1$; for option 1, $p = 1$, and for option 3, $p = 0.9$, both of which lead to divergence.

Which of the Three Series Below Converges? 1) ∑n=1∞1n\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }∑n=1∞​n1​

Which of the Three Series Below Converges?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$