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Suppose a power series converges to a function. Then, the even part of the power series converges to the even part of the function, and the odd part of the power series converges to the odd part of the function.
Explicitly, if:
f ( x ) = ∑ k = 0 ∞ a k x k {\displaystyle f(x)=\sum _{k=0}^{\infty }a_{k}x^{k}}
Then:
f ( x ) + f ( − x ) 2 = ∑ k = 0 ∞ a 2 k x 2 k {\displaystyle {\frac {f(x)+f(-x)}{2}}=\sum _{k=0}^{\infty }a_{2k}x^{2k}}
and:
f ( x ) − f ( − x ) 2 = ∑ k = 0 ∞ a 2 k + 1 x 2 k + 1 {\displaystyle {\frac {f(x)-f(-x)}{2}}=\sum _{k=0}^{\infty }a_{2k+1}x^{2k+1}}
