Can Term by Term Integration Give a Wrong Answer for Integrable Functions?

[elfboy]

to be integratable but still give a wrong answer even if all the steps are performed correctly? Assume that an infinite series is equal to some integral. Then you evaluate the integral between two values. Beforehand, you ensure that the series converges for those two endpoints. But the resulting answer is wrong even though the steps are correct.

 

[HallsofIvy]

NO, if a function is integrable, and you follow all of the steps involved in finding that integral correctly, you must get the correct value of the integral. However, finding the integral of an infinite sum by integrating term by term and then taking the limit may NOT be "performing the steps correctly"! That's your real question isn't it?

If the series converges uniformly then term by term integration is valid. There is a special situation: if a series of functions converges, then it converges uniformly in any compact (for sets of real number- closed and bounded) set. Since a finite interval, [a, b], is both closed and bounded, you can always integrate a series term by term from a to b as long as both a and b are finite. For an improper integral that may not be true.