e as a Limit

We can prove that the sequenceis monotone increasing and bounded. Therefore it has a limit. The limit is in fact

We can use the binomial theorem to write


henceis bounded above by 3

The kth term in the numerator is

The kth term in the denominator is

We must provethenand sequence is increasing.

The left hand side is greater than or equal tobut the left hand side is less than 3, sois increasing and bounded, hence converges.

To find the limit, take natural logs:

As both numerator and denominator tend to zero so use L'Hopital's Rule.

Exponentiation now givesas the limit of