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級數和問題02

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級數和問題02

請各位大大幫幫忙,要如何求出下面的級數和?謝謝!
\( \displaystyle \sum_{k=1}^n \frac{k+4}{k(k+1)(k+2)}=\frac{3}{2}-\frac{n+3}{(n+1)(n+2)} \)
\( \displaystyle \sum_{k=1}^{\infty} \frac{k+4}{k(k+1)(k+2)}=\frac{3}{2} \)

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回復 1# shmilyho 的帖子

\(\begin{align}
  & \frac{k+4}{k\left( k+1 \right)\left( k+2 \right)} \\
& =\frac{k+2+2}{k\left( k+1 \right)\left( k+2 \right)} \\
& =\frac{k+2}{k\left( k+1 \right)\left( k+2 \right)}+\frac{2}{k\left( k+1 \right)\left( k+2 \right)} \\
& =\frac{1}{k\left( k+1 \right)}+\frac{2}{k\left( k+1 \right)\left( k+2 \right)} \\
& =\left( \frac{1}{k}-\frac{1}{k+1} \right)+\left( \frac{1}{k\left( k+1 \right)}-\frac{1}{\left( k+1 \right)\left( k+2 \right)} \right) \\
\end{align}\)

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