Interval of Convergence of Partial Fractions
We can take the radius of convergence of the fraction
to be
- there are actually lots of radii of this sort, each corresponding to a different series expansion that represents the fraction– but that is for another time. Complications arise when we have the sum of two or more partial fraction since all the partial fractions may not be defined for all the partial fractions simultaneously. For example, we cannot substitute
into the expression
since the second term is not defined for
The first of these two expressions is defined for
and the second term is defined for
The intersection of these two intervals is actually just the second interval, and it is actually often the case that the overall interval of convergence is actually the intersection of all the separate intervals if they can be suitably expressed, and is often the smallest of all the intervals if they are taken around
as in the example above.
Example: Find the interval of convergence of the partial fractions
The interval of convergence of the first term is
and the interval of convergence of the second term is
The smallest of these two intervals is the second hence the interval of convergence of the whole expression is
The above technique only works when the denominator can be expressed into linear fraction. If a factor in the denominator has no real roots then we exclude it from these calculations. If a factor has real roots then these may need to be found so the smallest of all the intervals of convergence of all the linear terms and hence the overall interval of convergence can be written down.