Power Series
In this lesson, we explore the concept of power series, starting with the definition of a function defined by a power series. We see how a power series can be viewed as the limit of a sequence of polynomials, and how closed forms for series can be derived from geometric series. The radius of convergence theorem is also introduced, which provides us with a way to determine the domain of a power series. Finally, we explore the differentiation and integration of power series using power rules.
Functions defined by power series. Ratio and root tests for absolute convergence. Differentiation and integration. Closed forms for series derived from geometric series. Series expansions of ln(1+x) and inverse tan (x).
Copyright 2005, Department of Mathematics, University of Houston. Created by Selwyn Hollis. Find more information on videos, resources, and lessons at http://online.math.uh.edu/HoustonACT/videocalculus/index.html.
- What is a power series?
- What is a power series about x0?
- What are variations on the geometric series?
- How can you find closed forms for power series?
- What is a radius of convergence?
- How do you find the radius of convergence of a power series?
- What is the interval of convergence?
- How do you find the interval of convergence for a power series?
- How do you take the derivative and interval of a power series?
- What is a power series expansion?
- What is the power series expansion of ln(1 + x)?
- What is the power series expansion of inverse tan x?
- What is the exact value of the sum of k^2/2^k?
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Staff Review
Power series are defined and explained in depth in this lesson, as we are prepared for the final two lectures. The radius of convergence and interval of convergence are also defined, explained, and found for several different cases. Power series expansions are also found for several different functions. -
lgt9
The Houston videos are outstanding. I'm finding them really helpful for calculus 2 test preparation.
