Rewriting e as an infinite series

I'm given the function and I'm supposed to prove that .

I was thinking that I should use a different representation of e, namely:

and use the binomial theorem to expand it as an infinite series, but I can't get it to work properly. If it's possible, then how should it be done?

Otherwise, any hints on how I should prove it instead?

Re: Rewriting e as an infinite series

Quote:

Originally Posted by

**scounged** I'm given the function

and I'm supposed to prove that

.

The answer to this depends upon the level of rigor required in *proof*.

There is one well respected calculus textbook simply points out that in this case.

Re: Rewriting e as an infinite series

Quote:

Originally Posted by

**Plato** There is one well respected calculus textbook simply points out that

in this case.

Does that mean that is the only function that equals its own derivative?

If that's the case, then I think that's the way I'm intended to prove it (in a previous part of this assignment I'm supposed to prove that f(x)=f'(x).

Re: Rewriting e as an infinite series

Quote:

Originally Posted by

**scounged** Does that mean that

is the only function that equals its own derivative?

If that's the case, then I think that's the way I'm intended to prove it (in a previous part of this assignment I'm supposed to prove that f(x)=f'(x).

Well there is a trivial function, . But clearly that is not the case here.

Re: Rewriting e as an infinite series

Then I think I'll ask my teacher tomorrow if proving f(x)=f'(x) is enough. Thanks for the help.

Re: Rewriting e as an infinite series

Quote:

Originally Posted by

**scounged** I'm given the function

and I'm supposed to prove that

.

I was thinking that I should use a different representation of e, namely:

and use the binomial theorem to expand it as an infinite series, but I can't get it to work properly. If it's possible, then how should it be done?

Otherwise, any hints on how I should prove it instead?

Instead of using the Binomial Theorem, expand as a MacLaurin series.

Re: Rewriting e as an infinite series

Quote:

Does that mean that is the only function that equals its own derivative?

Quote:

Originally Posted by

**Plato** Well there is a trivial function,

. But clearly that is not the case here.

Not quite. , for A any constant (which, of course, includes 1 and 0), has that property.

Re: Rewriting e as an infinite series

Yeah, I found it in my formula collection. If I had known about MacLaurin series yesterday I don't think I would've created this thread. Anyway, thanks for the help.