求极限 $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\frac{i}{2^i}$.
求极限
\[
\lim_{n\rightarrow\infty}\frac{1}{n}(\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n})
\]
求极限
\[
\lim_{n\rightarrow\infty}\frac{1}{n}(\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n})
\]
1
令
\[
S_n=\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n},
\]
则
\[
2S_n=1+\frac{2}{2}+\frac{3}{2^2}+\frac{4}{2^3}+\cdots+\frac{n}{2^{n-1}},
\]
两式相减, 得
\[
\begin{split}
S_n&=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots+\frac{1}{2^{n-1}}-\frac{n}{2^n}\\
&=\frac{1-(\frac{1}{2})^n}{1-\frac{1}{2}}-\frac{n}{2^n}.
\end{split}
\]
于是
\[
\lim_{n\rightarrow\infty}\frac{1}{n}S_n=\lim_{n\rightarrow\infty}\frac{1}{n}\biggl[2(1-(\frac{1}{2})^n)-\frac{n}{2^n}\biggr]=0.
\]