Reminding students about Taylor series (Part 5)

Sadly, at least at my university, Taylor series is the topic that is least retained by students years after taking Calculus II. They can remember the rules for integration and differentiation, but their command of Taylor series seems to slip through the cracks. In my opinion, the reason for this lack of retention is completely understandable from a student’s perspective: Taylor series is usually the last topic covered in a semester, and so students learn them quickly for the final and quickly forget about them as soon as the final is over.

Of course, when I need to use Taylor series in an advanced course but my students have completely forgotten this prerequisite knowledge, I have to get them up to speed as soon as possible. Here’s the sequence that I use to accomplish this task. Covering this sequence usually takes me about 30 minutes of class time.

I should emphasize that I present this sequence in an inquiry-based format: I ask leading questions of my students so that the answers of my students are driving the lecture. In other words, I don’t ask my students to simply take dictation. It’s a little hard to describe a question-and-answer format in a blog, but I’ll attempt to do this below.

In the previous posts, I described how I lead students to the definition of the Maclaurin series

f(x) = \displaystyle \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!} x^k,

which converges to f(x) within some radius of convergence for all functions that commonly appear in the secondary mathematics curriculum.

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Step 5. That was easy; let’s try another one. Now let’s try f(x) = \displaystyle \frac{1}{1-x} = (1-x)^{-1}.

What’s f(0)? Plugging in, we find f(x) = \displaystyle \frac{1}{1-0} = 1.

Next, to find f'(0), we first find f'(x). Using the Chain Rule, we find f'(x) = -(1-x)^{-2} \cdot (-1) = \displaystyle \frac{1}{(1-x)^2}, so that f'(0) = 1.

Next, we differentiate again: f'(x) = (-2) \cdot (1-x)^{-3} \cdot (-1) = \displaystyle \frac{2}{(1-x)^3}, so that f''(0) = 2.

Hmmm… no obvious pattern yet… so let’s keep going.

For the next term, f'''(x) = (-3) \cdot 2(1-x)^{-4} \cdot (-1) = \displaystyle \frac{6}{(1-x)^4}, so that f'''(0) = 6.

For the next term, f^{(4)}(x) = (-4) \cdot 6(1-x)^{-5} \cdot (-1) = \displaystyle \frac{24}{(1-x)^5}, so that f^{(4)}(0) = 24.

Oohh… it’s the factorials again! It looks like f^{(n)}(0) = n!, and this can be formally proved by induction.

Plugging into the series, we find that

\displaystyle \frac{1}{1-x} = \sum_{n=0}^\infty \frac{n!}{n!} x^n = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + \dots.

Like the series for e^x, this series converges quickest for x \approx 0. Unlike the series for e^x, this series does not converge for all real numbers. As can be checked with the Ratio Test, this series only converges if |x| < 1.

The right-hand side is a special kind of series typically discussed in precalculus. (Students often pause at this point, because most of them have forgotten this too.) It is an infinite geometric series whose first term is $1$ and common ratio $x$. So starting from the right-hand side, one can obtain the left-hand side using the formula

a + ar + ar^2 + ar^3 + \dots = \displaystyle \frac{a}{1-r}

by letting a=1 and $r=x$. Also, as stated in precalculus, this series only converges if the common ratio satisfies $|r| < 1$, as before.

In other words, in precalculus, we start with the geometric series and end with the function. With Taylor series, we start with the function and end with the series.

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Step 6. A whole bunch of other Taylor series can be quickly obtained from the one for \displaystyle \frac{1}{1-x}. Let’s take the derivative of both sides (and ignore the fact that one should prove that differentiating this infinite series term by term is permissible). Since

\displaystyle \frac{d}{dx} \left( \frac{1}{1-x} \right) = \frac{1}{(1-x)^2}

and

\displaystyle \frac{d}{dx} \left( 1 + x + x^2 + x^3 + x^4 + \dots \right) = 1 + 2x + 3x^2 + 4x^3 + \dots,

we have

\displaystyle \frac{1}{(1-x)^2} = 1 + 2x + 3x^2 + 4x^3 + \dots.

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Next, let’s replace x with -x in the Taylor series in Step 5, obtaining

\displaystyle \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 \dots

Now let’s take the indefinite integral of both sides:

\displaystyle \int \frac{dx}{1+x} = \int \left( 1 - x + x^2 - x^3 + x^4 - x^5 \dots \right) \, dx

\ln(1+x) = \displaystyle x - \frac{x^2}{2} + \frac{x^3}{3} -\frac{ x^4}{4} + \frac{x^5}{5} -\frac{ x^6}{6} \dots + C

To solve for the constant of integration, let x = 0:

\ln(1) = 0+ C \Longrightarrow C = 0

Plugging back in, we conclude that

\ln(1+x) = x - \displaystyle \frac{x^2}{2} + \frac{x^3}{3} -\frac{ x^4}{4} + \frac{x^5}{5} -\frac{ x^6}{6} \dots

The Taylor series expansion for \ln(1-x) can be found by replacing x with -x:

\ln(1-x) = -x - \displaystyle \frac{x^2}{2} - \frac{x^3}{3} -\frac{ x^4}{4} - \frac{x^5}{5} -\frac{ x^6}{6} \dots

Subtracting, we find

\ln(1+x) - \ln(1-x) = \ln \displaystyle \left( \frac{1+x}{1-x} \right) = 2x + \frac{2x^3}{3}+ \frac{2x^5}{5} \dots

My understanding is that this latter series is used by calculators when computing logarithms.

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Next, let’s replace x with -x^2 in the Taylor series in Step 5, obtaining

\displaystyle \frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 + x^8 - x^{10} \dots

Now let’s take the indefinite integral of both sides:

\displaystyle \int \frac{dx}{1+x^2} = \int \left(1 - x^2 + x^4 - x^6 + x^8 - x^{10} \dots\right) \, dx

\tan^{-1}x = \displaystyle x - \frac{x^3}{3} + \frac{x^5}{5} -\frac{ x^7}{7} + \frac{x^9}{9} -\frac{ x^{11}}{11} \dots + C

To solve for the constant of integration, let x = 0:

\tan^{-1}(1) = 0+ C \Longrightarrow C = 0

Plugging back in, we conclude that

\tan^{-1}x = \displaystyle x - \frac{x^3}{3} + \frac{x^5}{5} -\frac{ x^7}{7} + \frac{x^9}{9} -\frac{ x^{11}}{11} \dots

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In summary, a whole bunch of Taylor series can be extracted quite quickly by differentiating and integrating from a simple infinite geometric series. I’m a firm believer in minimizing the number of formulas that I should memorize. Any time I personally need any of the above series, I’ll quickly use the above steps to derive them from that of \displaystyle \frac{1}{1-x}.

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7 Comments

  1. Square roots and logarithms without a calculator (Part 8) | Mean Green Math
  2. Formula for an infinite geometric series (Part 11) | Mean Green Math
  3. Fun lecture on geometric series (Part 2): Ways of counting money | Mean Green Math
  4. Is there an easy function without an easy Taylor series expansion? | Mean Green Math
  5. Reminding students about Taylor series: Index | Mean Green Math
  6. Thoughts on Infinity (Part 3g) | Mean Green Math
  7. Lessons from teaching gifted elementary students (Part 6b) | Mean Green Math

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