Power Series of Some Common Analytic Functions

Appendix C Power Series of Some Common Analytic Functions

We list the power series representation about 0 of some common analytic functions and indicate their intervals of convergence.

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\begin{aligned} \frac{1}{1 - x} & = \sum_{n = 0}^{\infty} x^{n} = 1 + x + x^{2} + x^{3} + \dots (- 1 < x < 1) \\ \ln (1 + x) & = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{n + 1}}{n + 1} = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \dots (- 1 < x \leq 1) \\ \arctan (x) & = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n + 1}}{2 n + 1} = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \dots (- 1 \leq x \leq 1) \\ e^{x} & = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots (- \infty < x < \infty) \\ \sin (x) & = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n + 1}}{(2 n + 1)!} = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots (- \infty < x < \infty) \\ \cos (x) & = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n}}{(2 n)!} = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots (- \infty < x < \infty) \end{aligned}

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Hit the "Evaluate" button below, you will see the function and its

k

th Taylor polynomial centered at the origin. Get an idea of how these approximations look like by varying the parameters.

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@interact
def taylor_approx(func=sin(x), order=5 , r=3):
    center = 0
    gfunc = plot(func,x,-r,r);
    gpoly = plot(taylor(func,x,center,order),x,-r,r,color='red');
    show(gfunc + gpoly,figsize=[4,2])

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