Editing Power series (section)

=== Addition and subtraction ===
When two functions ''f'' and ''g'' are decomposed into power series around the same center ''c'', the power series of the sum or difference of the functions can be obtained by termwise addition and subtraction. That is, if
<math display="block">f(x) = \sum_{n=0}^\infty a_n (x - c)^n</math> and <math display="block">g(x) = \sum_{n=0}^\infty b_n (x - c)^n</math>
then
<math display="block">f(x) \pm g(x) = \sum_{n=0}^\infty (a_n \pm b_n) (x - c)^n.</math>

The sum of two power series will have a radius of convergence of at least the smaller of the two radii of convergence of the two series,<ref>Erwin Kreyszig, Advanced Engineering Mathematics, 8th ed, page 747</ref> but possibly larger than either of the two. For instance it is not true that if two power series <math display="inline">\sum_{n=0}^\infty a_n x^n</math> and <math display="inline">\sum_{n=0}^\infty b_n x^n</math> have the same radius of convergence, then <math display="inline">\sum_{n=0}^\infty \left(a_n + b_n\right) x^n</math> also has this radius of convergence: if <math display="inline">a_n = (-1)^n</math> and <math display="inline">b_n = (-1)^{n+1} \left(1 - \frac{1}{3^n}\right)</math>, for instance, then both series have the same radius of convergence of 1, but the series <math display="inline">\sum_{n=0}^\infty \left(a_n + b_n\right) x^n = \sum_{n=0}^\infty \frac{(-1)^n}{3^n} x^n</math> has a radius of convergence of 3.