无穷级数例子
计算 lim x → ∞ ( 1 n + 1 + 1 n + 2 + 1 n + 3 + . . . + 1 n + 2 n − 1 + 1 n + 2 n ) 计算\lim _{x\to \infty} (\frac{1}{n+1} + \frac{1}{n+2}+\frac{1}{n+3} + ...+ \frac{1}{n+2n-1} + \frac{1}{n+2n} ) 计算x→∞lim(n+11+n+21+n+31+...+n+2n−11+n+2n1)
解：
lim x → ∞ ( 1 n + 1 + 1 n + 2 + 1 n + 3 + . . . + 1 n + 2 n − 1 + 1 n + 2 n ) = lim x → ∞ 1 2 n ( 1 1 2 + 1 2 n + 1 1 2 + 2 2 n + 1 1 2 + 3 2 n + . . . + 1 1 2 + 2 n − 1 2 n + 1 1 2 + 2 n 2 n ) = ∫ o 1 1 1 2 + x d x = ln ( 1 / 2 + x ) ∣ 0 1 = ln 3 − ln 2 − ln 1 + ln 2 = ln 3 \lim _{x\to \infty} (\frac{1}{n+1} + \frac{1}{n+2}+\frac{1}{n+3} + ...+ \frac{1}{n+2n-1} + \frac{1}{n+2n} )=\\ \lim _{x\to \infty} \frac{1}{2n}(\frac{1}{\frac{1}{2}+\frac{1}{2n}} + \frac{1}{\frac{1}{2}+\frac{2}{2n}}+\frac{1}{\frac{1}{2}+\frac{3}{2n}} + ...+ \frac{1}{\frac{1}{2}+\frac{2n-1}{2n}} + \frac{1}{\frac{1}{2}+\frac{2n}{2n}} )=\\ \int_o^1 \frac{1}{\frac{1}{2}+ x} dx = \ln(1/2+x)|_0^1 = \ln3 - \ln2 -\ln1 + \ln2 = \ln3 x→∞lim(n+11+n+21+n+31+...+n+2n−11+n+2n1)=x→∞lim2n1(21+2n11+21+2n21+21+2n31+...+21+2n2n−11+21+2n2n1)=∫o121+x1dx=ln(1/2+x)∣01=ln3−ln2−ln1+ln2=ln3