## 3.5. Special Sequences

### Definition 3.5.6: Euler Sequence

**Euler's Sequence**: Converges to

*e ~ 2.71828182845904523536028747135...*(Euler's number). This sequence serves to define

*e*.

**Euler's sequence**

### Proof:

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

Next, we need to show that the sequence is bounded. Again, consider the expansion- 0

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

1 +

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

so that, finally,

But then, putting everything together, we have shown thatfor alln.

for all1 + 1 + S_{n}3

*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