## Decreasing sequence Anastasia

Posted January 20th, 2010 by Structure

We want to prove that we have

or equivalent

We shall prove that sequence

has the property

As

we shall have

We also know that sequence is increasing and upper bounded and its limit is Euler number e.

So we also have

Now we return to sequence to prove that it is decreasing.

We have to prove

Or

This is equivalent to

or

But we have for all a>0

So it is enough to prove that

which is equivalent to

or

We have also

and so