Calculus I · Adding Up Forever

Power series: where Taylor’s expansions actually live

Now the payoff Chapter 7 has been owed. A power series $\displaystyle\sum_{n=0}^\infty c_n x^n$ is a Taylor series viewed as a sum that may or may not converge depending on $x$. The fundamental fact is clean:

Radius of convergence. A power series $\sum c_n x^n$ converges for all $|x| < R$ and diverges for all $|x| > R$, for some number $R$ (possibly $0$ or $\infty$) called the radius of convergence.

So every Taylor series comes with an interval $(-R, R)$ — centered at its center — inside which it genuinely equals its function, and outside which it is meaningless. This is the rigor Chapter 7 promised and postponed. And the practical tool for finding $R$ is the ratio test: form the ratio of consecutive terms and see what it approaches. (We introduce it here, where it actually matters — for power series — rather than as one more entry in a catalog.) If $\left|\dfrac{c_{n+1}}{c_n}\right| \to L$, then $R = \dfrac1L$, because the series behaves geometrically with ratio $\approx L|x|$, which converges precisely when $L|x| < 1$.

Turn it on Chapter 7’s flagship series:

• $e^x = \displaystyle\sum \frac{x^n}{n!}$: the ratio of coefficients is $\dfrac{1/(n+1)!}{1/n!} = \dfrac{1}{n+1} \to 0$, so $L = 0$ and $R = \infty$. The series converges for every $x$ — confirming, rigorously, the claim you proved by remainder in Chapter 7.
• $\dfrac{1}{1-x} = \displaystyle\sum x^n$: the coefficients are all $1$, so $L = 1$ and $R = 1$. Converges for $|x| < 1$, diverges beyond — that is the wall at $\pm 1$ in Chapter 7’s convergence figure, now derived rather than drawn.
• $\ln(1+x)$ and $\arctan x$: both have $R = 1$, so each is valid only on $(-1, 1)$ — which is why setting $x = 1$ in the arctangent series to reach $\tfrac{\pi}{4}$ was living dangerously, right at the boundary (it works, but only barely, and for a subtler reason).

So Chapter 7’s series were never just formal manipulations: each carries a precise domain of validity, and you can now compute it. (Figure 12.4: a power series’ interval of convergence drawn on the $x$-axis — solid inside $(-R, R)$, hatched outside — for $e^x$ (the whole line) and $\tfrac{1}{1-x}$ (a segment ending at $\pm1$).)

A last word: when alternating signs rescue a sum

One pattern deserves a closing mention, because Chapter 7’s sine, cosine, and arctangent series all have it: the terms alternate in sign. Alternation helps enormously, because consecutive terms partly cancel. The principle: if the terms alternate in sign and shrink steadily to zero, the series converges — even when the same terms with all-positive signs would diverge. The cleanest example is the alternating harmonic series $1 - \tfrac12 + \tfrac13 - \tfrac14 + \cdots$, which converges (to $\ln 2$, as the Chapter 7 series for $\ln(1+x)$ at $x=1$ suggests) — although, strip the signs and it becomes the harmonic series, which diverges. Cancellation is a real force. We won’t drill it into a formal apparatus; the idea — alternation tames a sum — is what to carry forward.

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