Answer
Diverges
Work Step by Step
The given series can be summed up as: $\dfrac{1}{2}+\dfrac{2}{5}+\dfrac{3}{17}+.....=\Sigma_{n=1}^{\infty} \dfrac{n}{n^2+1}$
We can see that the degree of the denominator is equal to $k=2$ and the degree of the numerator is equal to $j=1$.
This implies that $j \gt k$
Also, the condition $ j=1 \geq k-1=1$ has been satisfied.
Hence, we can conclude that the given series diverges.