The Taylor Series of the Exponential Function

Sorry for the lack of a post yesterday, but I was really tired after this weekend.

So what functions might we try finding a power series expansion for? Polynomials would be boring, because they already are power series that cut off after a finite number of terms. What other interesting functions do we have?

Well, one that’s particularly nice is the exponential function $\exp$ . We know that this function is its own derivative, and so it has infinitely many derivatives. In particular, $\exp(0)=1$ , $\exp'(0)=1$ , $\exp''(0)=1$ , …, $\exp^{(n)}(0)=1$ , and so on.

So we can construct the Taylor series at ${0}$ . The coefficient formula tells us

$\displaystyle a_k=\frac{\exp^{(k)}(0)}{k!}=\frac{1}{k!}$

which gives us the series

$\displaystyle\sum\limits_{k=0}^\infty\frac{x^k}{k!}$

We use the ratio test to calculate the radius of convergence. We calculate

$\displaystyle\limsup\limits_{k\rightarrow\infty}\left|\frac{\frac{x^{k+1}}{(k+1)!}}{\frac{x^k}{k!}}\right|=\limsup\limits_{k\rightarrow\infty}\left|\frac{x^{k+1}k!}{x^k(k+1)!}\right|=\limsup\limits_{k\rightarrow\infty}\left|\frac{x}{(k+1)}\right|=0$

Thus the series converges absolutely no matter what value we pick for $x$ . The radius of convergence is thus infinite, and the series converges everywhere.

But does this series converge back to the exponential function? Taylor’s Theorem tells us that

$\displaystyle\exp(x)=\left(\sum\limits_{k=0}^n\frac{x^k}{k!}\right)+R_n(x)$

where there is some $\xi_n$ between ${0}$ and $x$ so that $R_n(x)=\frac{\exp(\xi_n)x^n}{(n+1)!}$ .

Now the derivative of $\exp$ is $\exp$ again, and $\exp$ takes only positive values. And so we know that $\exp$ is everywhere increasing. What does this mean? Well, if $x\leq0$ then $\xi_n\leq0$ , and so $\exp(\xi_n)\leq\exp(0)=1$ . On the other hand if $x\geq0$ then $\xi_n\leq nx$ , and so $\exp(\xi_n)\leq\exp(x)$ . Either way, we have some uniform bound $M$ on $\exp(\xi_n)$ no matter what the $\xi_n$ are.

So now we know $R_n(x)\leq\frac{Mx^n}{(n+1)!}$ . And it’s not too hard to see (though I can’t seem to find it in my archives) that $n!$ grows much faster than $x^n$ for any fixed $x$ . Basically, the idea is that each time you’re multiplying by $\frac{x}{n+1}$ , which eventually gets less than and stays less than one. The upshot is that the remainder term $R_n(x)$ must converge to ${0}$ for any fixed $x$ , and so the series indeed converges to the function $\exp(x)$ .

About these ads

October 7, 2008 - Posted by John Armstrong | Analysis, Calculus, Power Series

5 Comments »

Is there a typo in the penultimate line? (gets less than and stays less than 1)

Comment by Richard Roe | October 8, 2008 | Reply
You’re right, sorry.

Comment by John Armstrong | October 8, 2008 | Reply
[...] now we know the Taylor series of the exponential function at [...]

Pingback by The Exponential Series « The Unapologetic Mathematician | October 8, 2008 | Reply
[...] we showed that it satisfied the exponential property. Now we’ve got another definition, using a power series, which is its Taylor series at . And we’ve shown that this definition also satisfies the [...]

Pingback by The Exponential Differential Equation « The Unapologetic Mathematician | October 10, 2008 | Reply
[...] this case, we can define . How does this work? we use the power series expansion of the [...]

Pingback by Automorphisms of Lie Algebras « The Unapologetic Mathematician | August 18, 2012 | Reply

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

RSS Feeds

RSS - Posts
RSS - Comments
Feedback

Got something to say? Anonymous questions, comments, and suggestions at Formspring.me!
Subjects
Archives

October 2008

M T W T F S S

« Sep Nov »

1 2 3 4 5

6 7 8 9 10 11 12

13 14 15 16 17 18 19

20 21 22 23 24 25 26

27 28 29 30 31
Search for:

The Unapologetic Mathematician

Mathematics for the interested outsider