Euler's Sequence: Converges toe ~ 2.71828182845904523536028747135...(Euler's number). This sequence serves to definee.(http://www.shu.edu/html/teaching/math/reals/numseq/speseq.html)back

**Euler's sequence**

We will show that the sequence is monotone increasing and bounded
above. If that was true, then it must converge. Its limit, by definition,
will be called *e* for Euler's number.

Euler's number *e* is irrational (in fact transcendental),
and an approximation of *e* to 30 decimals is *e ~ 2.71828182845904523536028747135*.

First, we can use the binomial theorem to expand the expression

Similarly, we can replace *n* by *n+1* in this expression
to obtain

The first expression has *(n+1)* terms, the second expression
has * (n+2)* terms. Each of the first *(n+1)* terms of
the second expression is greater than or equal to each of the *(n+1)* terms
of the first expression, because

But then the sequence is monotone increasing, because we have shown that

- 0

Next, we need to show that the sequence is bounded. Again, consider the expansion

1 +

Now we need to estimate the expression to finish the proof.

If we define *S _{n} = *,
then

so that, finally,

for all n.

But then, putting everything together, we have shown that

1 + 1 + S_{n}3

for all *n*. Hence, Euler's sequence is bounded by 3 for
all *n*.

Therefore, since the sequence is monotone increasing and bounded, it must converge. We already know that the limit is less than or equal to 3. In fact, the limit is approximately equal to 2.71828182845904523536028747135