Why am I not getting 1/e?? Instead I get e?
I am so stressed as I thought I had finally found the answer. Does anyone see the flaw in my solution? I don't see where I'm going wrong.
Help would be so apreciated.
I am not at all sure what exactly you are asking. If it is to derive the limit \(\displaystyle \mathop {\lim }\limits_{n \to \infty } {\left( {\frac{n}{{n + 1}}} \right)^n} = \frac{1}{e}\) or to prove it?
Here is a general principal: \(\displaystyle \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{a}{{n + b}}} \right)^{cn}} = {e^{ac}}\).
If we have proven (or were given) that rule then note that \(\displaystyle \left( {\frac{n}{{n + 1}}} \right) = \left( {1 + \frac{{ - 1}}{{n + 1}}} \right)\),
hence \(\displaystyle \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{{ - 1}}{{n + 1}}} \right)^n} = {e^{ - 1}}\) (i.e. \(\displaystyle a=-1\)).
If you are asking to derive the principal then that requires an entire section of an advanced text.