Substitution method for power series summation
Contents
Description of the method
The substitution method for power series summation is a method that can be used to convert one power series summation problem into another one. It is typically done with the goal of making the summation easier to obtain an explicit closed-form expression for.
Scalar multiple substitution
This is a substitution of the form for
a constant. Explicitly, consider a power series of the form:
(Note that the starting point could be 0, 1, or anything).
Then, with the substitution , this becomes:
We can use scalar multiple substitutions in order to get rid of purely exponential parts of the coefficients.
Power substitution
This is a substitution of the form for
a constant. Explicitly, consider a power series of the form:
(Note that the starting point could be 0, 1, or anything).
Then, with the subtsitution , this becomes:
Scalar multiple of power substitution
This combines the previous two substitution ideas, with a substitution of the form for
constants. Explicitly, consider a power series of the form:
(Note that the starting point could be 0, 1, or anything).
Then, with the subtsitution , this becomes:
Application
Goal of the substitution
The substitution method is typically used for two purposes:
- Get rid of unnecessary multiplicative exponential terms in the coefficients (the scalar multiple part takes care of this)
- Try to scale the exponent so that it better matches the coefficients (the power part takes care of this): The general rule is that, at the end of the substitution, the exponent should match, as closely as possible, any term that is in a denominator or whose factorial is in the denominator.
Combination with multiplication
Substitution can be combined with another common technique for power series manipulation: multiply and divide by in order to make the exponent better match the coefficient.
Related methods
- Exponent shift method for power series summation
- Integration and differentiation method for power series summation
Examples
Simple examples
Power series in ![]() |
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Scalar multiple or power or both? | New power series in ![]() |
Sum in term of ![]() |
Sum in terms of ![]() |
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power | ![]() |
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power | ![]() |
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scalar multiple | ![]() |
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combined | ![]() |
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power | ![]() |
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Examples that involve some combination of substitution and multiplying/dividing
Power series in ![]() |
manipulation + ![]() |
Scalar multiple or power or both? | New power series in ![]() |
Sum in term of ![]() |
Sum in terms of ![]() |
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first pull out a factor of ![]() ![]() |
power | ![]() |
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first multiply/divide by ![]() ![]() ![]() |
power | ![]() |
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Example involving a fractional power substitution
Consider the power series summation problem:
We want to do a -substitution that makes the exponent
so as to match the denominator. In order to do this, we would need to put
. This, however, is problematic since we don't know the sign of
. Thus, we make cases:
Case on sign of ![]() |
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Power series in terms of ![]() |
Sum in terms of ![]() |
Sum in terms of ![]() |
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positive | ![]() ![]() |
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negative | ![]() ![]() |
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At , either description fits. Overall, we have a piecewise definition of function for the sum of the series:
Interval of validity of power series summation
Transforming the interval of validity along with the substitution
Consider the -substitution
, giving:
We use some formula to sum up the power series rewritten in terms of . In the typical scenario, the formula we use is valid over the entire interval of convergence of
, though we may sometimes have a situation where the formula we've obtained is valid only on part of the interval. We then plug back
and get a formula for the summation in terms of
.
The final formula is valid precisely for those for which
is in the set of values of
for which the formula in terms of
is valid.
In the good case that the formula is valid over the entire interval of convergence for the power series in , we have:
- If the power series in
converges everywhere to the function we have obtained, then the power series in
also converges everywhere to the function we have obtained.
- If the radius of convergence for the power series in
is
, the radius of convergence of the power series in
is
.
- To determine whether the endpoints are included, apply the map
to the values
and check whether the image of the map is in the interval of convergence for
.
Let's now consider some of the finite cases more explicitly:
Sign of ![]() |
Parity of ![]() |
What can we say? |
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positive | odd | The negative endpoint ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
positive | even | Both endpoints ![]() ![]() ![]() ![]() |
negative | odd | The negative endpoint ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
negative | even | Both endpoints ![]() ![]() ![]() ![]() |
Multiplication and division
If the substitution method is combined with multiplying/dividing by a power of , then we also need to worry about the case
. In this case, the power series may still converge, but the formal expression we obtained may not be valid, though its limit would still be the right value.