derive e^x proof

I'm stuck on the first part!!
It's one of those things that is really obvious, but I can't think how to express it in a proof

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Reply 1

razzor

Differentiate the power series for e^x and see what you get!
Do you know what a factorial is?

Reply 2

TSR George

Original post by razzor

Differentiate the power series for e^x and see what you get!
Do you know what a factorial is?

Yep, it's just to prove it is true for all powers, I'd have to go through each one - I don't know how to express it by rule rather than by example, if that makes sense

Reply 3

Zacken

Do you mean you want to write

\displaystyle e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \Rightarrow \frac{\mathrm{d}}{\mathrm{d}x}(e^x) = \sum_{n=0}^{\infty} \frac{\mathrm{d}}{\mathrm{d}x} \left(\frac{x^n}{n!}\right) = \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x

Reply 4

an_atheist

Original post by Zacken

Do you mean you want to write

\displaystyle e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \Rightarrow \frac{\mathrm{d}}{\mathrm{d}x}(e^x) = \sum_{n=0}^{\infty} \frac{\mathrm{d}}{\mathrm{d}x} \left(\frac{x^n}{n!}\right) = \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x

I haven't reached this point yet, but is that it? It feels like there should be more to it.

Reply 5

Zacken

Original post by an_atheist

I haven't reached this point yet, but is that it? It feels like there should be more to it.

Nope, that's all there is to it - should take less than one line.

(of course, that's only at this level - in an undergraduate analysis course, you'd be expected to justify termwise differentiation (in this case, the series for e^x converges in an infinite radius), absolute convergence, swapping sum and operator, etc...)

(edited 7 years ago)

Reply 6

HapaxOromenon3

Original post by an_atheist

I haven't reached this point yet, but is that it? It feels like there should be more to it.

What Zacken wrote is correct, since d/dx(x^n / n!) = nx^(n-1) / n! = x^(n-1) / (n-1)!, so it's the same series, as can be seen by writing out the terms. In general you are allowed to differentiate a power series term by term as long as you stay within the radius of convergence.

Reply 7

TSR George

Original post by HapaxOromenon3

d/dx(x^n / n!) = nx^(n-1) / n! = x^(n-1) / (n-1)!

I got to this, but I wasn't sure whether this was technically correct as it would output different numbers with the same value of n as the original equation?

Reply 8

Zacken

Elaborate/give an example? What you've currently written makes no sense atm

Reply 9

TSR George

Original post by Zacken

Elaborate/give an example? What you've currently written makes no sense atm

So for when n= 2, using the e^x = 1 + x + ... equation, then the term would be 0.5x^2 but when using x^(n-1) / (n-1)! it would give x.

Because the term that n produces is different, can you still say that the expressions are the same?

Reply 10

HapaxOromenon3

Ultimately all the same terms get produced, just with the values of n shifted by 1.

Reply 11

TSR George

Original post by HapaxOromenon3

Ultimately all the same terms get produced, just with the values of n shifted by 1.

Exactly what I had thought but I wanted to make sure it was valid

Reply 12

Zacken

It's valid because there are an infinite number of terms.

Just like why taking the set of all integers and subtracting one from all of them would give you the set of all integers back.

that is:

f \colon \mathbb{Z} \to \mathbb{Z} \, \, \, \, n \mapsto n-1

is a bijective function.

Reply 13

TSR George

Original post by Zacken

It's valid because there are an infinite number of terms..

Ah I wondered if that might be the case. Ta

Reply 14

Zacken

No worries. Is that the Cambridge workbook for NatSci/CompSci or something? Are you a NatSci/CompSci fresher?

Reply 15

TSR George

Original post by Zacken

No worries. Is that the Cambridge workbook for NatSci/CompSci or something? Are you a NatSci/CompSci fresher?

It is! I'm afraid my brain has turned to mush over the summer :bawling:

I'm starting (Biological) NatSci this year!

Reply 16

Zacken

Oh wow! That's a lot of maths for Bio NatSci - congratulations! Never knew you were going to Camb. :smile:

Reply 17

TSR George

Original post by Zacken

Oh wow! That's a lot of maths for Bio NatSci - congratulations! Never knew you were going to Camb. :smile:

Haha I thought it would be a good idea for me to brush up before I go! Btw you may remember helping me do a load of M2 questions before the hols? (I had to self teach and there were loads of mistakes in the book) ... I got 100 UMS :biggrin: