8

I have figured out a way to show that

$$e=\sum_{i=0}^\infty {1\over i!}$$

I am wanting to formally show that

$$e^x = \sum_{i=0}^\infty {x^i\over i!}$$

I have been looking at power series/Taylor series for a long period of time (absolute convergence) and have seen multiple proofs that I look past because something seems illegitimate with radius of convergence. If someone explains the general proof behind Taylor series/power series absolutely converging so there is no gray area, that would work as well. I will probably ask some questions in the comment area if this is the case. There must be something I am missing.

Adam Hughes
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W. G.
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  • What exactly "seems illegitimate with radius of convergence?" If you want the convergence of Taylor series though, you need only read the proof of Taylor's theorem. – Adam Hughes Feb 09 '17 at 14:11
  • If you're requiring the radius of convergence for $e^x$ you need only use the Ratio test on the coefficients of the Taylor expansion. –  Feb 09 '17 at 14:13
  • You need to read the proofs for the ratio test and the root test. Then you can understand the radius of convergence. Both tests are based in the behaviour of a geometric series. –  Feb 09 '17 at 14:48
  • But knowing $e$, how do you define $e^x$ for real $x$ ? –  Feb 09 '17 at 19:47

3 Answers3

6

We have an amazing thing called Lagrange remainders. They basically tell us the difference between our function and it's Taylor polynomial. In general, we have

$$R_n(x)=|f(x)-P_n(x)|$$

where $P_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k$. Since it follows that

$$R_n(a)=0\\R_n'(a)=0\\R_n''(a)=0\\\vdots\\R_n^{(n)}(a)=0\\R_n^{(n+1)}(a)=|f^{(n+1)}(a)|$$

Thus,

$$R_n^{(n+1)}(x)\le|f^{(n+1)}(c)|$$

for some $c$ in our radius of convergence. It thus follows by integrating a few times that

$$R_n(x)\le\left|\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}\right|$$

One can then see that as $n\to\infty$, we have

$$|f(x)-P(x)|\le\lim_{n\to\infty}\left|\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}\right|$$

and if $\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}\to0$ for any $x,c$ within the a given domain, then the power series will equal the original function over that domain.

See if you can show that for any $x,c\in\mathbb R$,

$$\lim_{n\to\infty}\left|\frac{e^c}{(n+1)!}x^{n+1}\right|=0$$

On a side note, Lagrange remainder also shows us how well we approximate something when using a power series. For example, if I wanted to calculate $e$ out 5 places accurately,

$$R_n(x)=\left|e^x-\sum_{k=0}^n\frac{x^n}{n!}\right|\le0.000001$$

It's easy enough to solve, since

$$R_n(x)\le\left|e\frac{x^{n+1}}{(n+1)!}\right|\le\left|3\frac{x^{n+1}}{(n+1)!}\right|$$

Our particular case is $x=1$, and thus it suffices to solve

$$\frac3{(n+1)!}<0.000001$$

Which is easily done with a few checks to give $n\le8$. Thus,

$$e=\pm0.000001+\sum_{k=0}^8\frac1{k!}$$

Simply Beautiful Art
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    I have some problems with your answer. First, the absolute value in the definition of $R_n$ seems ro conflict with its usage later. Second, the phrase "integrating a few times" is to casual and shoyld be more explicit since this is the heart of the method. – marty cohen Feb 09 '17 at 14:54
  • I cannot deny, I don't fully remember how to derive Lagrange remainders. But I'm pretty sure my answer is the general outline. – Simply Beautiful Art Feb 09 '17 at 14:58
  • Why do u need to derive Lagrange remainder? It's part of statement of Taylor's theorem and can be taken for granted unless one specifically desires a proof of Taylor's theorem. If your objective is to apply Taylor's theorem then mentioning the right form of remainder is sufficient. – Paramanand Singh Feb 09 '17 at 18:57
  • You are a total boss! Nice answer! – W. G. Feb 09 '17 at 20:56
  • @W.G. :-) Your welcome. – Simply Beautiful Art Feb 09 '17 at 21:23
  • @ParamanandSingh Hm, good point. But I guess I thought it should be put in based on how I interpreted the question. – Simply Beautiful Art Feb 09 '17 at 21:24
5

You can do this in steps:

  1. Define the function $\exp x = \sum_{j=0}^\infty x^j/j!$. Show that it converges everywhere.

  2. Using Cauchy products and the Binomial Theorem, show that $\exp (w + z) = \exp w \exp z$.

  3. Show that $\exp$ is continuous and increasing. Differentiation gives $\exp' = \exp$, and $\exp$ is positive everywhere.

  4. Since you know $\exp 1 = \mathrm e$, an inductive application of (2) gives $$ (\exp q)^m = \exp(mq) = \exp n = \mathrm e^n $$ for any rational $q=n/m$, so $\exp q = \mathrm e^q$.

  5. Via the definition of power $\mathrm e^x = \sup_{q \leq x} \mathrm e^q$ and (3), show that $\exp x = \mathrm e^x$ for all $x$.

Henricus V.
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2

First two points about your question:

  • You say that you have figured out that $e=\sum 1/i!$. This suggests that you are using a definition of $e$ which is different from that sum. So how do you define $e$?
  • Next is how do you define $e^{x} $? If $x$ is rational then a definition of $e$ is sufficient. For irrational $x$ one must define the concept of irrational exponent and this is a non-trivial task. How do you do that?

There are many ways to answer both the questions above and the simplest way is perhaps to use the definition $$\exp(x)=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}\tag{1}$$ and then define $e=\exp(1)$. It can be proved that the definition above makes sense (that is the limit in $(1)$ exists). Further we can prove that $\exp(x+y) =\exp(x) \exp(y) $ and then by algebra we get $\exp(x) =e^{x} $ for all rational $x$.

Next step is to define irrational exponent $x$ at least for the base $e$ by the equation $e^{x} =\exp(x) $ and deal with symbol $a^{x}$ for general $a$ later.

The result $$\exp(x) =\sum_{i=0}^{\infty}\frac{x^{i}}{i!}\tag{2}$$ can be proved using $(1)$ with some effort. I have given details of this approach in my blog post.

If your objective is to establish $(2)$ using Taylor's theorem then you need to show that derivative of $\exp(x) $ is $\exp(x) $ itself. This is already presented in this answer.

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Paramanand Singh
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