Use the Direct Comparison Test (if possible) to determine whether the series $\sum _ { n = 5 } ^ { \infty } \frac { 1 } { 7 n ^ { 2 } + 4 }$ converges or diverges.
A) converges
B) diverges

Question

Quizplus · Accepted Answer

Note that $\frac{1}{7n^2+4}\leq\frac{1}{7n^2}$ for all $n\geq 5$. Therefore, $$\sum_{n=5}^\infty\frac{1}{7n^2+4}\leq\sum_{n=5}^\infty\frac{1}{7n^2}.$$ But we know that $\sum_{n=1}^\infty\frac{1}{n^2}$ converges, so $\sum_{n=1}^\infty\frac{1}{7n^2}$ converges by comparison. Hence, by the Direct Comparison Test, $\sum_{n=5}^\infty\frac{1}{7n^2+4}$ converges as well.

Use the Direct Comparison Test (If Possible) to Determine Whether ∑n=5∞17n2+4\sum _ { n = 5 } ^ { \infty } \frac { 1 } { 7 n ^ { 2 } + 4 }∑n=5∞​7n2+41​

Use the Direct Comparison Test (If Possible) to Determine Whether $\sum _ { n = 5 } ^ { \infty } \frac { 1 } { 7 n ^ { 2 } + 4 }$