These integrals turn up in subjects such as quantum field theory. Using your answer from (a), find a Taylor series for F(x) e-t2 dt centered at x 0. ![]() Write out the first six terms of this series don't bother to expand any factorials that show up. However, because the derivatives will not look. How do you use a Power Series to estimate the integral 0.01 0 sin(x2)dx Assuming that you know that the power series for sinx is: sinx n1 ( 1)n1 x2n1 2n 1 x x3 3 + x5 5 +. Then, you can substitute this into equation ( 1). ![]() ![]() Therefore, you can find the first 3 non-zero terms of the Taylor series by differentiating your function f ( x), and then substituting x 0 into those terms. The n + p = 0 mod 2 requirement is because the integral from −∞ to 0 contributes a factor of (−1) n+ p/2 to each term, while the integral from 0 to +∞ contributes a factor of 1/2 to each term. Question: Find a Taylor series representation for f(x) e-x2 centered at x 0. NB: I will assume that you want the Taylor Series centered at x 0 as on equation ( 1).
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