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1. I need to prove that (1 + x/n)^n tends to e^x as n tends to infinity. could someone give me a step by step proof?
2. (Original post by Shinjimonkey)
I need to prove that (1 + x/n)^n tends to e^x as n tends to infinity. could someone give me a step by step proof?
Do you know where to start? Have you tried anything?
3. (Original post by RDKGames)
Do you know where to start? Have you tried anything?
I am trying to:

ln(1+x/n)^n
= n ln (1+x/n)

but I do not know whether this is right or not
4. (Original post by Shinjimonkey)
I need to prove that (1 + x/n)^n tends to e^x as n tends to infinity. could someone give me a step by step proof?
We know that:

Can you work from there??
5. (Original post by RDKGames)
We know that:

Can you work from there??
how do we know the first rule that you have given?
6. (Original post by Shinjimonkey)
how do we know the first rule that you have given?
Because

Simple logarithm manipulation whereby
7. (Original post by RDKGames)
Because

Simple logarithm manipulation whereby
I see... as n tends to infinity, x/n tends to 0, so ln(1+x/n) tends to 0, therefore
n ln(1+x/n) = 0. then wouldn't (1+x/n)^n tend to 1?
8. (Original post by Shinjimonkey)
I see... as n tends to infinity, x/n tends to 0, so ln(1+x/n) tends to 0
Correct.

(Original post by Shinjimonkey)
therefore n ln(1+x/n) = 0. then wouldn't (1+x/n)^n tend to 1?
Not quite. Using the expansion for you would find that:

Therefore from the RHS you can see that as n goes to infinity, you are left with and not a 0.
9. (Original post by RDKGames)
Correct.

Not quite. Using the expansion for you would find that:

Therefore from the RHS you can see that as n goes to infinity, you are left with and not a 0.
I don't quite understand why ln(1+x/n) expands to that?
10. (Original post by Shinjimonkey)
I don't quite understand why ln(1+x/n) expands to that?
I hope you understand where the expansion for comes from.

If you're okay with that, the transition from to is quite straight forward as so you replace every with . Then of course, the whole series is simply multiplied by
11. (Original post by RDKGames)
I hope you understand where the expansion for comes from.

If you're okay with that, the transition from to is quite straight forward as so you replace every with . Then of course, the whole series is simply multiplied by
I do not quite understand why the expansion of ln(1+x) come from? Could you explain please? thank you for the help!
12. (Original post by RDKGames)
I hope you understand where the expansion for comes from.

If you're okay with that, the transition from to is quite straight forward as so you replace every with . Then of course, the whole series is simply multiplied by
I understand now. I just realised that ln(!+x) = integral of 1/(1+t) to the limits x to 0. Thanks for the help!
13. (Original post by Shinjimonkey)
I do not quite understand why the expansion of ln(1+x) come from? Could you explain please? thank you for the help!
It follows from Maclaurin's Series whereby any function can be rewritten as a sum of derivatives with increasing powers of x.

So let and perform this. You will get out with an infinite summation.

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