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So I'm very close to finishing a proof of the exponential function in terms of differential equations. For this next step, I have to show the following.

For $n \ge 0$ define $E_n (t)$ recursively according to $$\begin{gather} E_0 (t) = 1, \\ E_{n+1}(t) = 1 + \int _0 ^t E_n(s)ds. \end{gather}$$

Show the following: $\displaystyle E(t)= \sum_{n=0}^{+\infty} \frac{t^n}{n!}$.

Any help would be appreciated!

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Do you know that $E_n(t)$ is a convergent sequence?? –  B. S. Jan 23 '13 at 7:14
Or another way of putting this (sort of an intermediate step) is show that E_n(t) = 1 + t + t^2/2! + ... + t^n/n!. –  Bryce Jan 23 '13 at 7:16
yes, the sequence of partial sums converges according to the ratio test –  Bryce Jan 23 '13 at 7:16
I just had a TA help me. I was told E_0(t)=E_0(s)=1, which allows me to substitute this into my integral and get 1+t for E_1. Although I don't know why they're equivalent, this gives me the right solution –  Bryce Jan 23 '13 at 7:22
Problems involving recursion just beg for induction proofs... –  Antonio Vargas Jan 23 '13 at 7:50

2 Answers 2

We can do the problem by using Laplace transformation and also convolution of two functions.

As you noted, the sequence $E_{n+1}(t)=1+\int_0^{t}E_n(k)dk$ is convergent. So if $E_n(t)\to E(t)$ when $n\to\infty$ then we can have: $$E(t)=1+\int_0^{t}E(k)dk$$ According to what we knoe from convolution of two functions, we can write the latter identity as: $$E(t)=1+\int_0^{t}E(k)dk=1+1*E(t)$$ so we are led to the following ODE: $$E(t)=1+1*E(t)$$ Now take the Laplace of both sides and let $L(E(t))=Y$: $$Y=L(1+1*E(t))=\frac{1}{s}+\frac{1}{s}\times Y$$ So $Y=\frac{1}{s-1}$. This means that $E(t)=L^{-1}(\frac{1}{s-1})=\exp(t)$ .

Edit: Or, again we have : $$E(t)=1+\int_0^{t}E(k)dk$$ Differential from both sides, so: $$E'(t)=E(t)$$ a first order differential equation. It is separable and by solving it, you get $$\frac{d(E(t))}{E(t)}=dt$$ and so we have $E(t)=\exp(t)+C$. If you satisfy te condition you were given you will get $c=0$.

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Oh man, haha. We haven't learned Laplace transformations just yet. This is my first differential equations class. Also, thank you Arthur for editing it! –  Bryce Jan 23 '13 at 7:43
@Bryce: I added some notes. See them and I hope you get the answer. –  B. S. Jan 23 '13 at 8:59
I don't know why this has gone without an upvote for so long! +1 and well-deserved! –  amWhy Feb 10 '13 at 0:20
@amWhy: Thanks so much for your kind manner. :-) –  B. S. Feb 10 '13 at 3:30

We could use induction: For $E_1(t)$ we have $E_{1}(t)=1+t=\sum\limits_{n=0}^{1}\frac{t^n}{n!}$. Now assume that for some positive integer $k$ we have $$E_k(t)=\sum\limits_{n=0}^{k}\frac{t^n}{n!}$$ Then $$E_{k+1}(t)=1+\int\limits_{0}^{t} \sum\limits_{n=0}^{k}\frac{s^n}{n!}ds=1+ 1+\sum\limits_{n=0}^{k}\int\limits_{0}^{t} \frac{s^n}{n!}ds=1+\sum\limits_{n=0}^{k}\frac{t^{n+1}}{(n+1)n!}=1+\sum\limits_{n=1}^{k+1}\frac{t^n}{n!}=\sum\limits_{n=0}^{k+1}\frac{t^n}{n!}$$ By induction we now have $E_{j}(t)=\sum\limits_{n=0}^{j}\frac{t^n}{n!}$ and thus $E(t)=\sum\limits_{n=0}^{\infty}\frac{t^n}{n!}$.

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