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Advanced techniques

Integration using series

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

∫∑n=0∞anxn dx=∑n=0∞anxn+1n+1+C\int\sum_{n=0}^\infty a_nx^n\,dx=\sum_{n=0}^\infty\frac{a_nx^{n+1}}{n+1}+C

A power series can be integrated term by term inside its convergence radius. Other series require justification such as uniform convergence or domination. Endpoints need separate analysis.

Common mistake

Do not assume the radius automatically includes its endpoints.

Step-by-step example

  1. Use the geometric series on |x|<1.

    11−x=∑n=0∞xn\frac1{1-x}=\sum_{n=0}^\infty x^n

  2. Integrate from 0 to x inside that interval.

    ∫0xdt1−t=∑n=0∞xn+1n+1\int_0^x\frac{dt}{1-t}=\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}

  3. Relate the series to the logarithm.

    −ln⁡(1−x)=∑n=0∞xn+1n+1,∣x∣<1-\ln(1-x)=\sum_{n=0}^\infty\frac{x^{n+1}}{n+1},\quad|x|<1

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