<s>[INST] Limit of a infinite series $$\frac{2}{2} + \frac{2\cdot 5}{2\cdot 9} + \frac{2\cdot 5\cdot 10}{2\cdot 9\cdot 28} + \cdots + \frac{2\cdot 5\cdot 10 \cdots (n^2+1)}{2\cdot 9\cdot 28\cdots (n^3+1)}\tag1$$
For this series $(1)$, how would one go about applying the comparison test to check for convergence or divergence?
 [/INST] For $n\ge 3$ we get
$$\frac{n^2+1}{n^3+1}<\frac{n^2+1}{n^3}=\frac1n+\frac1{n^3}\le\frac13+\frac1{27}=\frac4{27}<\frac12$$
Then
$$\sum_{k=1}^n\prod_{j=1}^k\frac{j^2+1}{j^3+1}<1+\frac{5}{9}+\sum_{k=3}^n\frac59\left(\frac12\right)^{k-2}$$
So, by comparison test, we get that the given series converges.
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