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Showing exp(x) definitions are the same.

  1. Aug 27, 2013 #1
    1. The problem statement, all variables and given/known data
    [itex]e^{x}=\sum\limits_{k=0}^{\infty}\frac{x^{k}}{k!}[/itex] and [itex]e^{x}=\lim\limits_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}[/itex].

    I want to show that [itex]\sum\limits_{k=0}^{\infty}\frac{x^{k}}{k!}=\lim\limits_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}[/itex].

    2. The attempt at a solution
    Let $$t_{n}=\left(1+\frac{x}{n}\right)^{n}.$$ Then by the binomial theorem we have
    $$ t_{n}=1+x+\frac{x^{2}}{2!}\left(\frac{n-1}{n}\right)+\frac{x^{3}}{3!}\left(\frac{n-1}{n}\right)\left(\frac{n-2}{n}\right)+\cdots+\frac{x^{n}}{n!}\left(\frac{n-1}{n}\right)\left(\frac{n-2}{n}\right)\cdots\left(\frac{1}{n}\right)=\sum\limits_{k=0}^{n}\frac{x^{k}}{k!}\frac{n!}{n^{k}(n-k)!}.$$
    Then
    $$
    \begin{align*}
    \lim\limits_{n\rightarrow\infty}t_{n}=\lim\limits_{n\rightarrow\infty} \sum_{k=0}^{n}\frac{x^{k}}{k!}\frac{n!}{n^{k}(n-k)!}
    &=\sum_{k=0}^{\infty}\lim\limits_{n\rightarrow\infty}\frac{x^{k}}{k!}
    \frac{n!}{n^{k}(n-k)!}\\
    &=1+x+\frac{x^{2}}{2!}\lim\limits_{n\rightarrow\infty}\frac{n-1}{n}+\frac{x^{3}}{3!}\lim\limits_{n\rightarrow\infty}\frac{n-1}{n}\frac{n-2}{n}+\cdots \\
    &=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots \\
    &=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}.
    \end{align*}
    $$

    Hence
    $$\lim\limits_{n\rightarrow\infty}t_{n}=\lim\limits_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}=\sum\limits_{k=0}^{\infty}\frac{x^{k}}{k!}.$$

    Can someone check if my reasoning works? I am particularly worried about the part where I say
    $$
    \lim\limits_{n\rightarrow\infty} \sum_{k=0}^{n}\frac{x^{k}}{k!}\frac{n!}{n^{k}(n-k)!}=\sum_{k=0}^{\infty}\lim\limits_{n\rightarrow\infty}\frac{x^{k}}{k!}
    \frac{n!}{n^{k}(n-k)!}.
    $$
     
  2. jcsd
  3. Aug 27, 2013 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    That is certainly not true for arbitrary expressions.

    $$\lim\limits_{n\rightarrow\infty} \sum_{k=0}^{n} \frac{1}{n+1} = \lim\limits_{n\rightarrow\infty} 1 = 1$$
    $$\sum_{k=0}^{\infty}\lim\limits_{n\rightarrow\infty} \frac{1}{n+1} = \sum_{k=0}^{\infty} 0 = 0$$
     
  4. Aug 27, 2013 #3
    Hmm. Thanks. Could you recommend a way to tackle this problem?
     
  5. Aug 27, 2013 #4
    yeah putting the limit inside the summation requires some proving, because limit sum laws don't necessarily work for infinite sums. I think you could try something like that, for [itex]n ≥ 2 [/itex]:

    [tex]\left(1+\frac{x}{n}\right)^n = \sum_{k=0}^{n}\frac{n!}{k!(n-k)!}\frac{x^k}{n^k} = 1 + x + \sum_{k=2}^{n}\frac{n(n-1)...(n-k+1)}{n^k}\frac{x^k}{k!}[/tex]
    [tex] = 1 + x + \sum_{k=2}^{n}\left(1-\frac{1}{n}\right)...\left(1-\frac{k-1}{n}\right)\frac{x^k}{k!}[/tex].

    Then I am not quite sure how, but you'll need to prove that taking the limit inside the sum is ok. I tried some ideas but they failed miserably, so I'll think about it. It's an interesting problem, though!
     
    Last edited: Aug 28, 2013
  6. Aug 28, 2013 #5

    verty

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    Homework Helper

    I would try to expand the second definition for n = 1,2... and show how it relates to the first definition. Hopefully it correlates perfectly when n = k.

    Oh wait, it won't.
     
  7. Aug 28, 2013 #6

    Office_Shredder

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    Staff Emeritus
    Science Advisor
    Gold Member

    It's actually easier if you don't try to take both limits at the same time. The infinite summation
    [tex] \sum_{k=0}^{\infty} \frac{x^k}{k!} [/tex]
    is an infinite series that converges for all x. For each fixed n,
    [tex] \left(1+\frac{x}{n}\right)^n[/tex]
    is a power series (with only finitely many terms) that converges for all x. It's easier to show that as n goes to infinity, the latter converges to the former by taking their difference and noting that for each n, you have a power series that converges for each x - all that is left is to show that for a fixed x, as n goes to infinity the series goes to zero for any fixed x that you pick
     
  8. Aug 28, 2013 #7
    Interesting. It didn't cross my mind to think of it like that. Thank you.
     
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