- #1
Steve Turchin
- 11
- 0
Homework Statement
## lim_{n \rightarrow \infty}{\frac{1}{n^2} \sum_{k=1}^{n} ke^{\frac{k}{n}}} ##
Homework Equations
The Attempt at a Solution
## lim_{n \rightarrow \infty}{\frac{1}{n^2} \sum_{k=1}^{n} ke^{\frac{k}{n}}} \\
= lim_{n \rightarrow \infty}{\frac{1}{n^2} (1e^{\frac{1}{n}}+2e^{\frac{2}{n}}+3e^{\frac{3}{n}}+\ldots+ne^{\frac{n}{n}})} \\
= lim_{n \rightarrow \infty}{\frac{e^{\frac{1}{n}}}{n^2} +\frac{2e^{\frac{2}{n}}}{n^2} + \ldots + \frac{e}{n} } = 0 ##
I know the limit equals to 1 (Wolfram). Isn't the limit of a sum equals to the sum of the limits? What am I doing wrong?
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