- #1

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Using the fact that [tex]E_n=(1+\frac{1}{n})^n {\rightarrow}e[/tex], find the limits of the following:

a) [tex](1+\frac{1}{n^2})^{n^2}[/tex]

b)[tex](1+\frac{1}{2n})^n[/tex]

c)[tex](1+\frac{2}{n})^n[/tex]

Here is the work I've done:

a) If u=n

b) [tex](1+\frac{1}{2n})^n=((1+\frac{1}{2n})^{2n})^{\frac{1}{2}} {\rightarrow} e^{\frac{1}{2}}[/tex]

c) This is the part I'm having problems with. I'm having trouble manipulating the equation so that I can use the given fact about convergence to e. Any suggestions will be helpful.

Josh

a) [tex](1+\frac{1}{n^2})^{n^2}[/tex]

b)[tex](1+\frac{1}{2n})^n[/tex]

c)[tex](1+\frac{2}{n})^n[/tex]

Here is the work I've done:

a) If u=n

^{2}, [tex](1+\frac{1}{n^2})^{n^2}=(1+\frac{1}{u})^u {\rightarrow} e[/tex]b) [tex](1+\frac{1}{2n})^n=((1+\frac{1}{2n})^{2n})^{\frac{1}{2}} {\rightarrow} e^{\frac{1}{2}}[/tex]

c) This is the part I'm having problems with. I'm having trouble manipulating the equation so that I can use the given fact about convergence to e. Any suggestions will be helpful.

Josh

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