How can I prove that $\sum_n \left(\frac{n+1}{n^2+1}\right)^3$ converges?

Clearly $$\sum_n \left(\frac{n+1}{n^2+1}\right)^3$$ converges, but I am having an embarrassingly hard time proving it.

I tried the Cauchy Root test, leading me to prove that $$\limsup \left(\frac{n+1}{n^2+1} \right)^\frac{3}{n}=1$$, implying that the root test cannot be used. Since the ratio test is equivalent to the root test, I cannot use that either.

Any suggestions of directions I can take?