Evaluating Power Series

Now that we’ve got some topological fields to use as examples, let’s focus in on power series over $\mathbb{R}$ or $\mathbb{C}$ .

Remember that a power series is like an infinite polynomial. In fact, we introduced a topology so we could see in any power series a sequence of polynomials that converged to it. To be explicit, we write the series as a limit

$\displaystyle S=\sum\limits_{k=0}^\infty c_kX^k=\lim\limits_{n\rightarrow\infty}\sum\limits_{k=0}^nc_kX^k$

where the $c_k$ are coefficients selected from our base field.

Now evaluation of power series is specified by two conditions: it should agree with evaluation of polynomials when we’ve got a power series that cuts off after a finite number of terms, and it should be continuous.

The first condition says that each of our approximating polynomials should evaluate just the same as it did before. That is, if we cut off after the degree- $n$ term and evaluate at the point $x$ in the base field, we should get $\sum\limits_{k=0}^nc_kx^k$ .

The second condition says that evaluation should preserve limits. And we’ve got a sequence right here: the $n$ th term is the evaluation of the $n$ th approximating polynomial! So the power series should evaluate to the limit $\lim\limits_{n\rightarrow\infty}\sum\limits_{k=0}^nc_kx^k$ . If this limit exists, that is. And that’s why we need a topological field to make sense of evaluations.

Now we’re back in the realm of infinite series, and taking the limit of a sequence of partial sums. The series in question has as its $n$ th term the evaluated monomial $c_nx^n$ . We can start using our old techniques to sum these series.

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August 27, 2008 - Posted by John Armstrong | Analysis, Calculus, Power Series

4 Comments »

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