AP Calculus BC glossary

Term-by-term differentiation

Term-by-term differentiation means differentiating a power series one term at a time, as if it were a long polynomial. Inside the interval of convergence the result is the derivative of the sum, and it keeps the same radius of convergence, though behavior at the endpoints can change.

Differentiating 11−x=∑n=0∞xn\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n term by term gives 1(1−x)2=∑n=1∞nxn−1\frac{1}{(1-x)^2} = \sum_{n=1}^{\infty} n x^{n-1}, a quick way to build a new series without differentiating the closed form directly.

The radius of convergence stays fixed, but the starting index usually rises by one, since the derivative of the constant term is zero. Every other coefficient is scaled by its exponent.

The mistake

Assuming the interval of convergence carries over unchanged. The radius is preserved, but a convergent endpoint can drop to divergent once you differentiate, so each endpoint of the differentiated series has to be retested separately.

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