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Series expansions can be used to solve a wide range of equations including differential equations, functional equations, transcendental equations, and integral equations. Here are two examples.

Ideal Fermi Gas

The parametric form of the equation of state for a spinless ideal Fermi gas is

where can be determined in closed form.

The virial expansion of the equation of state is defined to be

where is called the th virial coefficient. Combining equations (1) and (2) yields

The virial coefficients can be obtained by direct expansion of equation (3) into a series in . Adding the term to an expression coerces it into a series up to order . For example,

shows that the leading term of is (in this case, obvious from the definition of ). Hence, to obtain the first four virial coefficients, we truncate the sum in equation (3) at and expand into a series up to order .

The virial coefficients, , are most easily extracted using .

Series Solution of Equations

In many situations, the summation symbol, , is not intended to imply explicit summation but is just a notational device. The Einstein summation convention, where summation over repeated indices is implied rather than explicitly stated, for example, can often be used to simplify the algebra. Interestingly, this convention translates into a general principle for analyzing and evaluating integrals and products of sums in Mathematica. Just focus on the summand (i.e., the general term of the sum) and drop the summation symbol.

For example, to determine the series solution to the differential equation , with initial condition , first substitute in the generic term of the series solution , which is .

The rule is used so that both and its derivatives are computed. Next, use pattern matching to find the recurrence relation satisfied by the .

This works by looking for the pattern (which means anything multiplied by where means anything or 0), and replacing it with where is reduced by and is omitted. This is, essentially, what a human does.

The solution to this recurrence relation can be obtained using the package. Alternatively, since

then determining in terms of ,

leads to general solution of the recurrence relation.

Here we have put , corresponding to the initial condition . Finally, the series solution, , is computed in closed form.

Of course, can solve this differential equation directly.

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