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Power Series: Formula & Examples

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  • 0:04 Recommendations & Suggestions
  • 1:06 Example: The Geometric Series
  • 1:57 Power Series: The Formula
  • 3:11 Lesson Summary
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Lesson Transcript
Instructor: Jasmine Cetrone

Jasmine has taught college Mathematics and Meteorology and has a master's degree in applied mathematics and atmospheric sciences.

A power series gives us what we call an infinite polynomial on our variable x and can be used to define a wide variety of functions. Learn the definition of power series as well as several examples of functions that can be described by them.

Recommendations & Suggestions

I'm sure you've used your calculator to help you calculate all sorts of answers in your mathematical career. Some things seem trivial, of course 3 * 4 = 12, but some are pretty amazing. How does it know that the natural logarithm of 1.3 is approximately 0.26236? Does it have a giant table of values that it looks up to give you the answer? That seems like it would require way too much storage space. So, how does it do it?

Turns out that many of the functions that we have a lot of trouble calculating by hand (trigonometric functions, logarithms, exponentials, etc.) can also be expressed using polynomials. Remember polynomial functions only have whole-numbered powers on our variable. Polynomials are generally very easy to calculate as they consist of just a bunch of additions and multiplications. The difference with some of our functions is that these polynomials have an infinite number of terms. This is what we call a power series.

Example: The Geometric Series

The simplest power series is the geometric series, and is expressed as:


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It is the sum of all powers of x from zero to infinity. Each of these powers of x has a coefficient of one. We can collapse this series into sigma notation.


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The geometric series is special in that it's one of the rare series that we actually have a formula for the sum. If x is between -1 and 1, the infinite sum is equal to 1/(1-x). What this also tells us is that the function 1/(1-x) can be expressed as a power series. Pretty neat that a simple function can be written as an infinite polynomial!


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Power Series: The Formula

How can we get different functions out of a power series? We can manipulate the polynomials in different ways:

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