The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { 3 ^ { k } + 5 ^ { k } } { 5 ^ { k } + 7 ^ { k } }$

Question

Quizplus · Accepted Answer

The terms of the series do not approach zero as $k$ approaches infinity, which is a necessary condition for convergence. Specifically, as $k$ increases, both $3^k + 5^k$ and $5^k + 7^k$ are dominated by their highest power terms, making the fraction approach a non-zero constant ($\frac{5^k}{7^k} \rightarrow \frac{1}{7^k}$ as $k \rightarrow \infty$, which does not approach 0).

The Given Series Converges ∑k=1∞3k+5k5k+7k\sum _ { k = 1 } ^ { \infty } \frac { 3 ^ { k } + 5 ^ { k } } { 5 ^ { k } + 7 ^ { k } }∑k=1∞​5k+7k3k+5k​

The Given Series Converges $\sum _ { k = 1 } ^ { \infty } \frac { 3 ^ { k } + 5 ^ { k } } { 5 ^ { k } + 7 ^ { k } }$