Exponent shift method for power series summation
Description of the method
This method is used for summing up various power series that look similar to existing power series but have some terms missing, extra, or shifted. It is closely related to another method, the substitution method for power series summation.
Missing terms in summation
Suppose we know how to do a certain summation , and we are asked to do a summation , where is a positive integer. This is given by:
Shifted index of summation
Suppose we know how to do a summation of the form:
Let's now consider another summation, with integers such that :
We simplify this as follows
We now use the missing terms in summation idea to simplify this final summation in terms of the original.
|Series we need to sum||Type||Rewrite in terms of known series||Final answer|
|Missing terms only|
|Shifted index of summation + missing terms||, expression valid for . At , we get a value of .|
|Shifted index of summation||for , 1 for .|