<s>[INST] Find partial sum of this series: $\sum_{n=1}^{\infty } \frac{2n+3}{n(n+1)(n+2)(n+3)}$? $\sum_{1}^{\infty } \frac{2n+3}{n(n+1)(n+2)(n+3)}$
How to find partial sum, sum and prove convergence by definition? Thanks a lot.
 [/INST] $$\sum_{n=1}^{\infty } \frac{2n+3}{n(n+1)(n+2)(n+3)}$$
Resolve it into partial fractions,
$$=\frac{1}{2}\sum_{n=1}^{\infty }[\frac{1}{n}-\frac{1}{n+1}-\frac{1}{n+2}+\frac{1}{n+3}]$$
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