The series in question is $$\sum_{n=1}^\infty\frac{n^2+1}{2n^2+5}$$
If $a_n$ is the $n$'th term of this sum, then $a_n \rightarrow \frac{1}{2}$ as $n\rightarrow \infty$. I believe this implies that the series doesn't converge – since this is effectively the negation of the statement "If a series converges, then $a_n$ tends to $0$ as $n$ tends to $\infty$". If I prove that the sequence $x_n=\frac{n^2 + 1}{2n^2 +5}$ converges to $\frac{1}{2}$, have I then proven that the series diverges? Or should I apply a convergence test and show that it doesn't pass one?
Best Answer
The infinite series $$ \sum_{n=1}^\infty\frac{n^2+1}{2n^2+5}$$ diverges because the term $$\frac{n^2+1}{2n^2+5}\to 1/2 \not=0$$
Intuitively you are adding infinitely many numbers which are very close to $1/2$ and the result does not converge.
The so called divergence test indicates that if the general term of a series does not tend to zero, then the series diverges.
You do not have to prove anything else for the divergence of the above series.