Power Series

General form of a Power Series

A power series is a series of the form

We call

the center of the series. It's best to think of a power series as a polynomial with an infinite number of terms. If we wrote it out, you can see this

In general, a power series is only convergent for a subset of the real numbers. One of the benefits of working with power series is that they can be integrated and differentiated term by term.

Many functions are differentiable at the origin, and since it's a simpler form, we frequently center the power series here to produce what is known as a Maclaurin series

We can play around with power series to find expressions for series of numbers that would otherwise be difficult to find.

Example: Use the Maclaurin series for to evaluate

Example: Evaluate

STEP 2012, Math III, #4: (i) Show that

and

Sum the series

(ii) Sum the series

We can also split a power series into two separate sigmas, one containing the even terms and the other containing the odd terms. Here is how it works.

be the coefficient of in the series expansion, in ascending powers of , of

where . Show, using partial fractions, that either or according to the value of .

[First use partial fractions to separate the expression into a sum of three distinct rational functions of the form

Now write each rational function as a power series. You can use the known power series

to find all three of them by substitution or differentiation.

One of these power series will have only even terms. You can split the other two series into two power series, one having only the even terms and the other having only odd terms using the method above.

So group all the even terms together into one sigma and all the odd terms together into another sigma. Simplify. You will now have formulas for the coefficients of the power series of the original function. Verify that they satisfy the conditions stated.]

Hence find a decimal approximation, to nine significant figures, for the fraction .

[You are not required to justify the accuracy of your approximation.]

STEP 2009, Math III, #3: The function is defined, for , by

, show that . Find and evaluate . [Yes, both of these limits could be done with L'Hopital's Rule, but the point of the exercise is to use power series.]

(ii) Show that is an even function. [Note: A function is said to be even if .]

(iii) Show with the aid of a sketch that and deduce that for .

Sketch the graph of .

STEP 2006, Math III, #3: (i) Let

and

for

. Explain why for even .

Prove the identity

and show that

.

(ii) Let

for . By considering , or otherwise, show that

.

(iii) Show that

.

Deduce from this and the previous results that

, , ... , and be any fixed real numbers. The numbers and are defined by

,

and the numbers

and are defined by

,

(i) Express

in terms of , and .

(ii) Show that

(iii) Express

in terms of , , and .

Hence show that for all values of , but that if and only if