Properties of Power Series

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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

and

then

It is not true that if two power series and have the same radius of convergence, then also has this radius of convergence. If and , then both series have the same radius of convergence of 1, but the series has a radius of convergence of 3.

Multiplication and division

With the same definitions for and , the power series of the product and quotient of the functions can be obtained as follows:

The sequence is known as the convolution of the sequences and .

For division, if one defines the sequence by

then

and one can solve recursively for the terms by comparing coefficients.

Solving the corresponding equations yields the formulae based on determinants of certain matrices of the coefficients of and

Differentiation and integration

Once a function is given as a power series as above, it is differentiable on the interior of the domain of convergence. It can be differentiated and integrated quite easily, by treating every term separately:

Both of these series have the same radius of convergence as the original one.

Resources

Differentiation and Integration of Power Series

Power Series Representation of Functions Using Differentiation and Integration

Licensing

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