If f(x) = integral [cos(t^2)dt] from 0 to x,
using the known power series for cos x, find a power series for f(1).

Question

If f(x) = integral [cos(t^2)dt] from 0 to x,
using the known power series for cos x, find a power series for f(1).

kirbykirby · Answer

This is f(x): $$f(x)=\int\limits_{0}^{X}\cos(t^2)dt$$

kirbykirby · Answer

I ended up finding $$\sum_{n=0}^{infinity}\frac{(-1)^{n}x^{4n+1}}{(4n+1)(2n)!}=x-\frac{x^5}{5*2!}+\frac{x^9}{9*4!}-\frac{x^{13}}{13*6!}+...$$

kirbykirby · Answer

But I'm not quite sure what it means to find the power series for f(1).. Isn't that just a number (whatever I found evaluated at 1)? There is no more x-dependence so how is it a power series :S I'm so confused

anonymous · Answer

Did you mean using known McLaurin series, or Taylor series for Cos (x)?

kirbykirby · Answer

No. This is the exact question, and the power series to find is for that function above (the integral one), although the question states to find it for f(1) specifically.