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opus
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Homework Statement
Determine the series that is equal to the integral ##\int_0^1 x^2\cos(x^3)dx##
Homework Equations
The Attempt at a Solution
So I didn't really know what I was doing but I did end up with the correct solution.
What I did was to find a Taylor Series for the integrand, this turned out to be ##\sum_{n=0}^\infty (-1)n\frac{x^{6n+2}}{(2n)!}##
Now I have ##\int_0^1\sum_{n=0}^\infty (-1)^n\frac{x^{6n+2}}{(2n)!}dx##
After some research, I found that for power series (for which a Taylor series is) you can put the integral inside the sum (if this is incorrect, please let me know).
Now I have ##\sum_{n=0}^\infty (-1)^n\frac{1}{(2n)!}\int_0^1 x^{6n+2}dx##
This looks like a mess, but not sure how I should have grouped this the proper way. That is, should I have put everything after ##\sum## in brackets?
Eventually, I end up evaluating and I get ##\sum_{n=0}^\infty \frac{(-1)^n}{(2n)!(6n+3)}##
Now normally, a series would never equal an integral to my understanding because series are for positive integers. But this is a power series, which involves x. So is it true that this is why the series can equal the integral in this case? Because x can take on ANY real positive values now?
I guess thinking about a series like a "bar graph" kind of thing isn't helping me much in trying to understand this.