# The Power of Taylor Series

I am teaching a Calculus class and we are finishing up power/Taylor series this week. The last section of the chapter is on applications, but the only ones listed there are approximating non-rational numbers like $\sqrt{1.02}$ and computing limits like $\lim_{x\to 0}\frac{\sin x}{x}$. I would like to find better examples that may or may not have a quick physical application (I cannot assume they know any physics beyond what I can explain). So, my question is, do any of you know some applications of Taylor series that I could spend maybe about half an hour to forty five minutes doing? They needn't be physical applications, just interesting. I have already done Euler's formula. Also, we do not deal with any remainder theorems in this class. Thanks.

• "Integration" of things like $e^{x^2}$? Oct 18 '11 at 19:15
• Solving differential equations.
Oct 18 '11 at 19:18
• Approximate a solution of $x^4 + y^4 = 2 x y$ near $(0,0)$ by $y = x^3/2 + x^{11}/32 + \ldots$. Oct 18 '11 at 19:31
• Related: Motivating infinite series. Oct 18 '11 at 19:52
• Taylor's theorem is used quite ingeniously in a proof of the central limit theorem (one doesn't need the whole series there, though. Just a few terms). Unfortunately, this probably doesn't fit into a 30-45 hour frame (unless the students have prior background in probability).
– Mark
Oct 18 '11 at 20:31

Not quite on the power of Taylor's series, but you could use the Taylor series for $f(x) = e^x$ to show that $e$ is irrational.

Here is an interesting application of power series; unfortunately one would need to bother with the remainder to make it really interesting.

$$\arctan(x)= \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{2n+1} \,.$$

Plug in $x= \frac{1}{\sqrt 3}$ ($1$ would also work but one would need to explain why this formula also holds at the end point of the interval). We get:

$$\frac{\pi}{6} = \frac{1}{\sqrt{3}}\sum_{n=0}^\infty \frac{(-1)^n}{3^n2n+1} \,.$$

The right side is an alternating series which converges very fast, thus you can use it to calculate $\pi$ with 5-6 digits. And it is alternating, which means you could use the Alternating Series error estimate.

You can also do the same for the Taylor series of $e^x$.

• Of course, some smart-ass will ask "how do you compute $\sqrt{3}$"? (This isn't hard - Newton's method converges quickly, for example, or one could find the series for $\sqrt{x}$ around $x=4$ - but it's worth thinking about beforehand.) Oct 18 '11 at 19:51
• Nice example. One can alternately use $\pi/4=\arctan(1/2)+\arctan(1/3)$. Oct 18 '11 at 19:54
• ...and in general, there's a whole family of Machin-type formulae. The trick of course for speedy convergence is that the arguments of the arctangent should be as small (that is, near the expansion point) as possible. Oct 18 '11 at 23:44

I presume the students would know how to sum (finite/infinite) geometric series. You can tell them that the infinite geometric series can be seen as the Taylor expansion of the function $\frac{1}{1-x}$: $$\frac{1}{1-x} = 1+ x + x^2 + x^3 + \cdots.$$ Integrating term by term (not always valid!), $$-\ln (1-x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \cdots,$$ which is nothing but the Taylor expansion of the function $\ln(1+x)$. (Of course, you can also derive the Taylor expansion directly, if you prefer.) Finally, plugging in $x=-1$ (once again, not really valid!), we get the sum of the alternating harmonic series: $$\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots.$$

There are many unjustified steps in the above derivation, but it gives a cool demonstration of power series and Taylor expansions.

• Nice examples. To compute $\ln2$, one should definitely use $x=\frac12$ instead of $x=-1$. The unjustified steps disappear and, perhaps even more importantly, the series one obtains converges much more quickly.
– Did
Nov 3 '11 at 8:23

You can show that a certain function $f$ does not have any zero between $-1$ and $1$, for instance, by showing that $1/f$ is analytic with a radius of analyticity at least $1$ (and therefore finite).