Lebl-Basic-Analysis-I · Problem 6.2.19

Sequences of Functions

Mathematics · power-series-integrated-radius-convergence

Problem

(requires §2.6) If ∑n=0∞cn(x−a)n\sum_{n=0}^\infty c_n(x-a)^n has radius of convergence ρ\rho, show that the term by term integral ∑n=1∞cn−1n(x−a)n\sum_{n=1}^\infty \dfrac{c_{n-1}}{n}(x-a)^n has radius of convergence ρ\rho. Note that we only proved above that the radius of convergence was at least ρ\rho.

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