easy convergence question

**worc3247** · Jan 4th 2011, 11:46 AM

This question seems like it should be so easy, but I just can't quite get there:

Use the fact that $(1 + \frac{1}{n})^n \rightarrow e$ to deduce that $(1 - \frac{1}{n})^n \rightarrow \frac{1}{e}$ .

So far this is what I have but I have a feeling i'm going about it the wrong way:
$(\frac{n+1}{n})^n \rightarrow e$
By algebra of limits: $(\frac{n}{n+1})^n \rightarrow \frac{1}{e}$
So $(1- \frac{1}{n+1})^n \rightarrow \frac{1}{e}$ .
But from here I can't see how I can get the answer required.

**snowtea** · Jan 4th 2011, 11:58 AM

Substitute $n = -m$ and simplify (if you know the limit holds for positive and negative n).

For your solution
$(1- \frac{1}{n+1})^n = (1- \frac{1}{n+1})^{n+1}(1 - \frac{1}{n+1})^{-1} = (1- \frac{1}{n+1})^{n+1}(\frac{n+1}{n})$
Now use the substitution $n = m - 1$

**worc3247** · Jan 4th 2011, 12:08 PM

Cheers

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