Thread: Converge or Diverges? View Single Post
 P: 280 $$\lim_{n\rightarrow \infty} \left (\left ( 1+\frac{1}{n} \right )^n \right)^2$$ $$\left ( \lim_{n\rightarrow \infty} \left ( 1+\frac{1}{n} \right )^n \right)^2$$ So let's just ignore the squaring for now. Whatever we find, we will square it. $$I =\lim_{n\rightarrow \infty} \left ( 1+\frac{1}{n} \right )^n$$ $$ln(I) =\lim_{n\rightarrow \infty} ln\left (\left ( 1+\frac{1}{n} \right )^n \right )$$ $$ln(I) =\lim_{n\rightarrow \infty} nln\left (1+\frac{1}{n} \right )$$ This is an indeterminate form of zero times infinity. Let $$N = \frac{1}{n}$$ then $$ln(I) =\lim_{n\rightarrow 0} \frac{ln\left (1+N \right )}{N}$$ Can you take it from here?