Home/Calculus II/Power series and Taylor series/Term-by-term differentiation and integrationSection 15.2Term-by-term differentiation and integrationWithin ∣x−a∣<R|x-a|<R∣x−a∣<R:Copy LaTeXddx∑n=0∞cn(x−a)n=∑n=1∞ncn(x−a)n−1\frac{d}{dx}\sum_{n=0}^{\infty}c_n(x-a)^n =\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}dxdn=0∑∞cn(x−a)n=n=1∑∞ncn(x−a)n−1View detail →FormulaCopy LaTeX∫∑n=0∞cn(x−a)n dx=C+∑n=0∞cnn+1(x−a)n+1.\int\sum_{n=0}^{\infty}c_n(x-a)^n\,dx =C+\sum_{n=0}^{\infty}\frac{c_n}{n+1}(x-a)^{n+1}.∫n=0∑∞cn(x−a)ndx=C+n=0∑∞n+1cn(x−a)n+1.View detail →The radius of convergence is preserved, although the endpoints may change.
Within ∣x−a∣<R|x-a|<R∣x−a∣<R:Copy LaTeXddx∑n=0∞cn(x−a)n=∑n=1∞ncn(x−a)n−1\frac{d}{dx}\sum_{n=0}^{\infty}c_n(x-a)^n =\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}dxdn=0∑∞cn(x−a)n=n=1∑∞ncn(x−a)n−1View detail →
FormulaCopy LaTeX∫∑n=0∞cn(x−a)n dx=C+∑n=0∞cnn+1(x−a)n+1.\int\sum_{n=0}^{\infty}c_n(x-a)^n\,dx =C+\sum_{n=0}^{\infty}\frac{c_n}{n+1}(x-a)^{n+1}.∫n=0∑∞cn(x−a)ndx=C+n=0∑∞n+1cn(x−a)n+1.View detail →