Review alternating-series questions by testing decreasing terms, the zero limit and whether convergence is absolute or conditional.
A function f has derivatives of all orders for -1<x<1. The derivatives of f satisfy the conditions above. The Maclaurin series for f converges to f(x) for |x|<1.
Determine whether the Maclaurin series described in part (a) converges absolutely, converges conditionally, or diverges at x=1. Explain your reasoning.
For x=1, the Maclaurin series becomes ∑n=1∞n(−1)n+1.
The series does not converge absolutely because the harmonic series diverges.
The series alternates with terms that decrease in magnitude to 0, and therefore the series converges conditionally.