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# derive e^x proof

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1. 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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2. (Original post by pippabethan)
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
Differentiate the power series for e^x and see what you get!
Do you know what a factorial is?
3. (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
4. (Original post by pippabethan)
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
Do you mean you want to write
5. (Original post by Zacken)
Do you mean you want to write
I haven't reached this point yet, but is that it? It feels like there should be more to it.
6. (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...)
7. (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.
8. (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?
9. (Original post by pippabethan)
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?
Elaborate/give an example? What you've currently written makes no sense atm
10. (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?
11. (Original post by pippabethan)
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?
Ultimately all the same terms get produced, just with the values of n shifted by 1.
12. (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
13. (Original post by pippabethan)
Exactly what I had thought but I wanted to make sure it was valid
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: is a bijective function.
14. (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
15. (Original post by pippabethan)
Ah I wondered if that might be the case. Ta
No worries. Is that the Cambridge workbook for NatSci/CompSci or something? Are you a NatSci/CompSci fresher?
16. (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
I'm starting (Biological) NatSci this year!
17. (Original post by pippabethan)
It is! I'm afraid my brain has turned to mush over the summer
I'm starting (Biological) NatSci this year!
Oh wow! That's a lot of maths for Bio NatSci - congratulations! Never knew you were going to Camb.
18. (Original post by Zacken)
Oh wow! That's a lot of maths for Bio NatSci - congratulations! Never knew you were going to Camb.
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
19. (Original post by pippabethan)
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
Yeah, there's never too much maths.

I do remember you very well, yes! That's amaaazing. Well done.
20. (Original post by Zacken)
Yeah, there's never too much maths.

I do remember you very well, yes! That's amaaazing. Well done.
Well thank you so much for the help!!

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