Answer

问题及解答

计算 $\sum_{k=1}^{n}\frac{1}{1+2+\cdots+k}$

Posted by haifeng on 2020-10-29 17:06:07 last update 2022-09-20 20:20:39 | Edit | Answers (1)

计算

\[1+\frac{1}{1+2}+\frac{1}{1+2+3}+\cdots+\frac{1}{1+2+3+\cdots+n}\]

 

并求极限

\[
\lim_{n\rightarrow\infty}\biggl(1+\frac{1}{1+2}+\frac{1}{1+2+3}+\cdots+\frac{1}{1+2+3+\cdots+n}\biggr)
\]

1

Posted by haifeng on 2022-09-25 12:59:40

\[
\frac{1}{1+2+\cdots+k}=\frac{1}{\frac{k(k+1)}{2}}=\frac{2}{k(k+1)}=2(\frac{1}{k}-\frac{1}{k+1}),
\]

因此

\[
\begin{split}
\sum_{k=1}^{n}\frac{1}{1+2+\cdots+k}&=\sum_{k=1}^{n}2(\frac{1}{k}-\frac{1}{k+1})\\
&=2\Bigl[(\frac{1}{1}-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+\cdots+(\frac{1}{n}-\frac{1}{n+1})\Bigr]\\
&=2(1-\frac{1}{n+1}).
\end{split}
\]

\[
\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{1+2+\cdots+k}=2.
\]