Derivative of a Power Series - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.01SC | Single Variable Calculus
Academic year:

2019
Views:

127

Pages:

1
Author:

Jovan Salas

Derivative of a Power Series We can diﬀerentiate power series. For example, cos(x) = sin� (x) so we can ﬁnd a power series for cos(x) by diﬀerentiating the power series for sin(x) term by term — the same way we diﬀerentiate polynomials. sin(x) = x − cos(x) cos(x) x3 x5 x7 + − + ··· 3! 5! 7! sin� (x) x2 x4 x6 = 1 − 3 + 5 − 7 + ··· 3! 5! 7! x2 x4 x6 = 1− + − + ··· 2! 4! 6! = x2 x2 Notice how 3 became when we canceled the 3’s. This happens with each 3! 2! term of the power series. The radius of convergence of the derivative of a power series is the same as the radius of convergence of the power series you started with. Here R = 1. Of course, you could get this same formula using Taylor’s formula and the derivatives of the cosine function.

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