Chapter 23 — Key Takeaways

A compact recap of Power Series and Taylor Series. Use it as a pre-exam checklist; every item links back to a section of the chapter.

1. Power series and the radius of convergence (Section 23.2)

A power series centered at $a$ is

$$\sum_{n=0}^\infty c_n (x - a)^n.$$

  • It converges on an interval centered at $a$, governed by the radius of convergence $R$.
  • Trichotomy: either $R = 0$ (converges only at $x = a$), $0 < R < \infty$ (converges absolutely for $|x-a| < R$, diverges for $|x-a| > R$), or $R = \infty$ (converges for all $x$).
  • Compute it with the ratio test (Chapter 22): $\;R = \displaystyle\lim_{n\to\infty}\left|\frac{c_n}{c_{n+1}}\right|\;$ when the limit exists; otherwise use Cauchy–Hadamard, $R = 1/\limsup_n |c_n|^{1/n}$.
  • The endpoints $|x-a| = R$ are never automatic. The ratio test is silent there ($\rho = 1$), so substitute $x = a \pm R$ and test each numerical series by hand.

2. Functions defined by power series (Section 23.3)

Inside the radius of convergence a power series defines a beautifully behaved function: it is continuous, and you may differentiate and integrate it term-by-term,

$$f'(x) = \sum_{n=1}^\infty n c_n (x-a)^{n-1}, \qquad \int f\,dx = C + \sum_{n=0}^\infty \frac{c_n}{n+1}(x-a)^{n+1},$$

with the same radius $R$ preserved (endpoint behavior may change).

3. Taylor and Maclaurin series (Section 23.4)

For a function with all derivatives at $a$, the Taylor series is forced by term-by-term differentiation:

$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x - a)^n.$$

  • Centered at $a = 0$, it is the Maclaurin series.
  • The degree-$N$ truncation is the Taylor polynomial $T_N$ — the unique degree-$\le N$ polynomial matching $f$ and its first $N$ derivatives at $a$. ($T_1$ is exactly the Chapter 11 tangent-line linearization.)

4. The seven series to memorize (Section 23.4)

Function Maclaurin series Interval
$e^x$ $\displaystyle\sum_{n=0}^\infty \dfrac{x^n}{n!}$ $(-\infty,\infty)$
$\sin x$ $\displaystyle\sum_{n=0}^\infty \dfrac{(-1)^n x^{2n+1}}{(2n+1)!}$ $(-\infty,\infty)$
$\cos x$ $\displaystyle\sum_{n=0}^\infty \dfrac{(-1)^n x^{2n}}{(2n)!}$ $(-\infty,\infty)$
$\dfrac{1}{1-x}$ $\displaystyle\sum_{n=0}^\infty x^n$ $(-1,1)$
$\ln(1+x)$ $\displaystyle\sum_{n=1}^\infty \dfrac{(-1)^{n+1} x^n}{n}$ $(-1,1]$
$\arctan x$ $\displaystyle\sum_{n=0}^\infty \dfrac{(-1)^n x^{2n+1}}{2n+1}$ $[-1,1]$
$(1+x)^k$ $\displaystyle\sum_{n=0}^\infty \binom{k}{n} x^n$ $(-1,1)$, endpoints depend on $k$

The binomial coefficient $\binom{k}{n} = \dfrac{k(k-1)\cdots(k-n+1)}{n!}$ is valid for any real $k$.

5. Taylor's theorem and the error bound (Section 23.5)

$$f(x) = T_N(x) + R_N(x), \qquad R_N(x) = \frac{f^{(N+1)}(c)}{(N+1)!}(x-a)^{N+1}$$

for some unknown $c$ between $a$ and $x$ (Lagrange form). We never need $c$ — only a bound. If $|f^{(N+1)}| \le M$ on the interval, then

$$\boxed{\;|R_N(x)| \le \frac{M}{(N+1)!}\,|x-a|^{N+1}\;}$$

This is the certificate that tells you, before computing, how many terms guarantee a target accuracy. For an alternating series, the simpler bound "error $\le$ first omitted term" (Chapter 22) often suffices.

6. Building new series from old (Section 23.6)

Four legal moves inside the radius of convergence — almost never the raw derivative formula:

  • Substitute an expression for $x$: $e^{-x^2} = \sum (-1)^n x^{2n}/n!$.
  • Differentiate term-by-term: $\sum n x^{n-1} = 1/(1-x)^2$.
  • Integrate term-by-term: integrating $1/(1+x^2)$ gives $\arctan x$; integrating $1/(1+x)$ gives $\ln(1+x)$.
  • Multiply via the Cauchy product $\left(\sum a_n x^n\right)\!\left(\sum b_n x^n\right) = \sum_n\!\left(\sum_{k=0}^n a_k b_{n-k}\right)x^n$.

7. The anchor payoff (Section 23.7)

Term-by-term integration computes the "impossible" integral:

$$\int_0^a e^{-x^2}\,dx = a - \frac{a^3}{3} + \frac{a^5}{10} - \frac{a^7}{42} + \cdots,$$

and likewise the normal CDF $\Phi(z)$, even though $e^{-x^2}$ has no elementary antiderivative. This closes the "area under the normal curve" anchor opened in Chapter 13 — the engine behind every z-table and erf call.

8. Why the radius is what it is (Sections 23.9–23.10)

  • The radius of convergence equals the distance from the center to the nearest singularity in the complex plane. That is why $1/(1+x^2)$ has $R = 1$ despite being smooth on all of $\mathbb{R}$: its poles sit at $\pm i$, distance $1$ away. And it is why $e^x, \sin x, \cos x$ (entire functions) have $R = \infty$.
  • Smooth is not the same as analytic. $f(x) = e^{-1/x^2}$ (with $f(0)=0$) has every derivative zero at $0$, so its Maclaurin series is identically $0$ — yet $f \ne 0$. A function equals its Taylor series only when the remainder $R_N \to 0$.

9. Common errors to avoid

  • Forgetting the endpoints. Computing $R$ and declaring an open interval, when the endpoints may converge ($\sum x^n/n^2$ at both) or split ($\sum x^n/n$ at one).
  • Dropping cross terms when multiplying series. The $x^4$ coefficient of $e^x\sin x$ needs every product of powers summing to 4 — the Cauchy product, not just the $x^4$ terms.
  • Confusing "has a Taylor series" with "equals its Taylor series." Always check that $R_N \to 0$ (the $e^{-1/x^2}$ trap).
  • Trusting the $e^{-x^2}$ series for large arguments. Catastrophic cancellation wrecks it; libraries switch to asymptotic expansions in the tail.

10. Connections

  • Back to Chapter 11: Taylor polynomials are linearization continued to every order; $T_1$ is the tangent line.
  • Back to Chapter 22: the ratio test fixes the radius; the alternating series test settles endpoints and bounds truncation error.
  • Back to Chapters 13–14: the normal-curve anchor and the Fundamental Theorem, finally resolved here.
  • Forward to Chapter 24: feeding $i$ into the series for $e^x$, $\sin x$, $\cos x$ yields Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ and the identity $e^{i\pi} + 1 = 0$ — the toolbox of this chapter spent on the masterpiece of the next.

Skills checklist

  • [ ] Find the radius and full interval of convergence (ratio test + endpoint checks).
  • [ ] Write any of the seven standard series from memory, and build others by substitution.
  • [ ] Compute a Taylor/Maclaurin series from the definition when needed.
  • [ ] Bound a Taylor-polynomial error with the Lagrange remainder, and solve for the number of terms.
  • [ ] Differentiate and integrate series term-by-term, including approximating $\int e^{-x^2}\,dx$.

Reflection

The deepest lesson of this chapter is the one that has run through the whole book: approximation is not a retreat from exactness but a road to it. A polynomial you can compute, plus a remainder you can bound, equals a transcendental value you can trust to as many digits as you please. Every sine your calculator returns, every small-angle approximation in physics, and every probability under the bell curve is an infinite polynomial, summed term by careful term. You now know how that quiet machinery works.