AP Calculus BC glossary

Term-by-term integration

Term-by-term integration means integrating a power series one term at a time. Inside the interval of convergence the result is an antiderivative of the sum, with the same radius of convergence. It is the standard way to derive the series for arctangent and for the natural log of one plus x.

Starting from $\frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n$ and integrating term by term gives $\ln(1+x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{n+1}}{n+1} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ .

The same move on $\frac{1}{1+x^2} = \sum_{n=0}^{\infty} (-1)^n x^{2n}$ produces $\arctan x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots$ , which is far faster than repeated differentiation.

The mistake

Forgetting the constant of integration. Integrating term by term introduces a $+C$ that you pin down at the center. For $\ln(1+x)$ , setting $x = 0$ gives $\ln 1 = 0$ , so $C = 0$ ; skip that and the whole series is off by a constant.

Appears in: Unit 10: Infinite Sequences and Series (BC)