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So, I know there are a few functions that don't have elementary antiderivatives, and the way I approach some of these problems is by integrating the Taylor series. In the process, I usually interchange the (indefinite) integral sign and the summation sign. Example: $$ \int \sin(x)/x\ dx\,=\\\int \sum_{n=0}^\infty (-1)^nx^{2n}/(2n+1)!\ dx=\\\sum_{n=0}^\infty ((-1)^{n}/(2n+1)!)\int x^{2n} dx=\\C+\sum_{n=0}^\infty (-1)^{n}x^{2n+1}/((2n+1)!(2n+1)) $$ Is this valid? when doing indefinite integrals of this sort (integrating Taylor series), when, if ever, will it be invalid to interchange the integration and summation signs?

clathratus
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    It's usually appropriate to make such maneuvers, as both Riemann Integration and Lesbegue Integration are consistent under such swappages. I personally find baby Rudin to be a good guide for this question, although any elementary analysis text should do the question justice. – Rushabh Mehta Aug 10 '18 at 02:15
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    A power series can be integrated term-by-term. That is, for Taylor series you can always interchange integration and summation with the circle of convergence. – saulspatz Aug 10 '18 at 02:28
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    Various answers here. – David Aug 10 '18 at 02:31

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