Theory

Theory 1

Given a numerical value for within the interval of convergence of a power series, the series sum may be considered as the output of a function .

Many techniques from algebra and calculus can be applied to such power series functions.

Addition and Subtraction:

Summation notation:

Scaling:

Summation notation:

Extra - Multiplication and composition

Multiplication:

For example, suppose that the geometric power series converges, so . Then we have for its square:

Composition:

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Assume:

Then:

Differentiation:

Antidifferentiation:

For example, for the geometric series we have:

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Do the series created with sums, products, derivatives etc., all converge? On what interval?

For the algebraic operations, the resulting power series will converge wherever both of the original series converge.

For calculus operations, the radius is preserved, but the endpoints are not necessarily:

Power series calculus - Radius preserved

If the power series has radius of convergence , then the power series and also have the same radius of convergence .

Power series calculus - Endpoints not preserved

It is possible that a power series converges at an endpoint of its interval of convergence, yet and do not converge at .

Extra - Proof of radius for derivative and integral series

Suppose has radius of convergence :

Consider now the derivative and its successive-term ratios:

Consider now the antiderivative and its successive-term ratios:

In both these cases the ratio test provides that the series converges when .