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## Homework Statement

Prove that the rate of convergence for Euler's method (in solving the problem y'=y) is comparable to 1/n by showing that [tex]lim_{n\rightarrow infinity} \frac{e-(1+\frac{1}{n})^n}{1/n}[/tex].

## Homework Equations

Hint: Use L'Hopital's Rule

If lim x-> infinity f(x)/g(x) is in indeterminate form, then it can be evaluated as lim x-> infinity f'(x)/g'(x).

Hint: Use Maclaurin expansion for ln(1+t).

ln(1+t) = t - t^2/2 + t^3/3 - t^4/4 + t^5/5 ...

## The Attempt at a Solution

L'Hopital's Rule:

Differentiating gives

[tex]lim_{n\rightarrow infinity} \frac{\frac{1}{n+1} - ln(1 + \frac{1}{n})(\frac{n+1}{n})^n}{-1/n^2}[/tex]

[tex] ln (1+1/n) = 1/n - 1/(2n^2) + 1/(3n^3) - 1/(4n^4) + 1/(5n^5) ... [/tex]

Not sure where to go from here. When I plug in the series, what can I do? I tried multiplying the fraction by [tex]\frac{-n^2}{-n^2}[/tex] but I don't think that gets me anywhere. I could apply L'Hopital again, but that seems like a monstrous function to differentiate. Any ideas?