NYJC/2021/JC2/Prelim/P1/Q03

(i) By first expressing $\frac{1}{r !(r+2)}$ in the form $\frac{A}{(r+2) !}+\frac{B}{(r+1) !}$ where $A$ and $B$ are constants, find $\sum_{r=1}^n \frac{1}{r !(r+2)}$

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(ii) Hence, explain why the series $\sum_{r=1}^{\infty} \frac{1+\left(\frac{1}{3}\right)^r r !(r+2)}{r !(r+2)}$ converges and find the exact sum to infinity of this series.

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