Determine if the series converges absolutely, converges, or diverges.
-$$\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 7 } { 4 } - \frac { 2 } { n } \right) ^ { n }$$
A) Diverges
B) converges conditionally
C) Converges absolutely

Question

Quizplus · Accepted Answer

The series diverges because the terms do not approach zero as $n$ approaches infinity. For a series to converge, a necessary condition is that the terms of the series must approach zero. In this case, the base of the exponent, $\frac{7}{4} - \frac{2}{n}$, approaches $\frac{7}{4}$ as $n$ goes to infinity, and since $\frac{7}{4} > 1$, raising it to the power of $n$ will cause the terms to grow without bound rather than approach zero.

Determine If the Series Converges Absolutely, Converges, or Diverges ∑n=1∞(−1)n(74−2n)n\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 7 } { 4 } - \frac { 2 } { n } \right) ^ { n }n=1∑∞​(−1)n(47​−n2​)n

Determine If the Series Converges Absolutely, Converges, or Diverges $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 7 } { 4 } - \frac { 2 } { n } \right) ^ { n }$