For which of the following series will the Ratio Test fail to give a definite answer (i.e., be inconclusive)?
1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 99 } { 100 } \right) ^ { n }$
2) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 100 } { 99 } \right) ^ { n }$
3) $\sum _ { n = 1 } ^ { \infty } n ^ { - 100 }$
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Question

For which of the following series will the Ratio Test fail to give a definite answer (i.e., be inconclusive)?
1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 99 } { 100 } \right) ^ { n }$ 
2) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 100 } { 99 } \right) ^ { n }$ 
3) $\sum _ { n = 1 } ^ { \infty } n ^ { - 100 }$ 
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 Ratio Test is inconclusive if the limit of the ratio of successive terms is 1. For series 1 and 2, the ratio of successive terms will not approach 1, but for series 3, which is a p-series with p > 1, the Ratio Test does not apply directly and can be inconclusive for determining convergence.

For Which of the Following Series Will the Ratio Test ∑n=1∞(99100)n\sum _ { n = 1 } ^ { \infty } \left( \frac { 99 } { 100 } \right) ^ { n }∑n=1∞​(10099​)n

For Which of the Following Series Will the Ratio Test $\sum _ { n = 1 } ^ { \infty } \left( \frac { 99 } { 100 } \right) ^ { n }$