Which of the following series converges?
1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }$
2) $\sum _ { n = 1 } ^ { \infty } 2 ^ { n }$
3) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + n ^ { 3 } }$
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Question

Which of the following series converges?
1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }$ 
2) $\sum _ { n = 1 } ^ { \infty } 2 ^ { n }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + n ^ { 3 } }$ 
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Quizplus · Accepted Answer

The third series converges because it is a p-series with $p > 1$ (since the denominator grows as $n^3$), which means it decreases sufficiently fast to ensure convergence. The first series does not converge because it oscillates between 1 and -1, and the second series diverges because it grows exponentially.

Which of the Following Series Converges? 1) ∑n=1∞(−1)n\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }∑n=1∞​(−1)n

Which of the Following Series Converges?
1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }$