#MathsInMinutes Day 49: Power series

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#MathsInMinutes Day 49: Power series

A power series is the sum of the terms in an ordered list, where those terms involve increasing positive powers of a variable x. The geometric progression

1 + x + x² + x³ + …

is a special case, in which the coefficients of each term are equal to 1. Power series are much more general than they might appear, and many functions can be written as power series. If all the coefficients beyond a given term to a zero, then the power series is finite and forms a polynomial.

Can power series converge? Using the theory of geometric progressions, we can tell that, if x is between -1 and 1, then the partial sum of the series above converges to 1/(1-x).

Of course not all power series obey such rules, but comparisons with simple geometric progressions can often be used to determine whether or not they do.

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