Determine whether the sequence convergent or divergent. $$\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ^ { 2 } + 3 }$$
A) converges
B) diverges

Question

Quizplus · Accepted Answer

We can use the comparison test to show that the sequence converges.

First, we note that $\frac{3}{n^2 + 3} < \frac{3}{n^2}$ for all $n \geq 1$ (since $n^2 + 3 > n^2$).

Also, we know that $\sum_{n=1}^\infty \frac{3}{n^2}$ converges since it is a p-series with $p = 2 > 1$.

Therefore, by the comparison test, we have that

$$0 \leq \sum_{n=1}^\infty \frac{3}{n^2 + 3} \leq \sum_{n=1}^\infty \frac{3}{n^2}$$

And since the outer two series converge, the middle series must also converge by the squeeze theorem.

Determine Whether the Sequence Convergent or Divergent ∑n=1∞3n2+3\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ^ { 2 } + 3 }n=1∑∞​n2+33​