In Exercises 41–50, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.
Fourier Series The function, ex, is called the fundamental periodic function as it is a combination of sin mx and cos mx. A periodic function, f(x), over [−π,π] can be expanded by a linear combination of the fundamental periodic functions as ∞ ∑ imx f(x) = cme . (1) m=−∞ −inx Multiplying e on the both sides of eq.(1) and integrating the result from −π to π yields ∫ ∞ ∫ ▯ −inx ∑ ▯ i(m−n)x f(x)e dx = cm e dx (2) −▯ m=−∞ −▯