Speaker A
Professor Dave here. Let’s check out power series. We’ve seen a few different types of series up until this point, so let’s check out another.
An introduction to power series, convergence tests, radius and interval of convergence, with examples and the ratio test explained.
Key Takeaways
- Power series can be represented as infinite polynomials with coefficients.
- Convergence depends on the values of x and can be tested using the ratio test.
- Radius of convergence defines the interval where the series converges.
- There are three main convergence scenarios: convergence at a single point, convergence everywhere, or convergence within a radius.
- Examples illustrate how to apply the ratio test and interpret results.
What the video covers
- Definition of power series as infinite sums of coefficients times powers of x.
- Explanation of coefficients and representation of power series as functions.
- Discussion of convergence and divergence of power series.
- Example of a geometric series with coefficients all equal to one and its convergence interval.
- Use of the ratio test to determine convergence of power series involving (x - a)^n terms.
- Derivation of radius of convergence and interval of convergence from the ratio test.
- Theorem describing three possible convergence cases for power series.
- Explanation of radius of convergence as zero, finite, or infinite depending on the series.
- Example using factorial terms to show infinite radius and interval of convergence.
- Summary of key concepts and encouragement to check comprehension.
Chapters
- 00:00Introduction to Power Series
- 00:23Terms and Coefficients of Power Series
- 00:34Function Representation and Infinite Polynomials
- 00:58Examples of Convergence and Divergence
- 01:52Geometric Series and Convergence Interval
- 02:11Using the Ratio Test for Convergence
- 02:39Simplifying the Ratio Test Expression
- 03:39Further Simplification and Limit Evaluation
- 04:17Determining the Interval of Convergence
- 05:20Theorem on Convergence Cases
Full Transcript — Download SRT & Markdown
Speaker A
A power series is one that takes the form C sub N times X to the N power, from zero to infinity.
Speaker A
So we get C zero first, with no X, because X to the zero is one.
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Then the next term is C one times X. Then C two times X squared, and so forth.
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The C’s are constants that we call coefficients, and the sum of this series can be represented by a function, F of X, where we just add up all the terms, resulting in what looks like a polynomial with infinitely many terms.
Speaker A
The domain of this function is the set of all X values for which this series converges.
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Now let’s look at some specific examples, and see what we can determine regarding convergence and divergence.
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Let’s say we take this general form and we make all the coefficients equal to one.
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That would give us this geometric series here, which will be convergent when X is in between negative one and one.
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We may also see power series in this form, where the binomial X minus A is being raised to the N power.
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How can we assess whether this kind of power series is convergent or divergent? Sometimes we will use the ratio test.
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Take something like the quantity X minus three raised to the N power, over N. Remembering the ratio test from the previous tutorial, let’s change N into N plus one, and then take the regular version, and flip it to bring it up here.
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To simplify, X minus three raised to the N plus one can become X minus three raised to the N times X minus three raised to the one.
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This more complicated term then cancels, leaving us with X minus three times N over N plus one.
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Now let’s divide this fraction by N, to get one over the quantity one plus (one over N).
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As these are all positive, we can pull them out of the absolute value brackets, leaving just X minus three in there.
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Now, bringing N up to infinity, this whole part just becomes one, leaving us with the absolute value of X minus three.
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In order to be convergent, this has to be less than one, so given the absolute value, we know that X minus three must be less than one and greater than negative one.
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Adding three throughout, we get X as being between two and four, so those are the values for which the series converges.
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In general, there is a theorem that summarizes the three possibilities that can describe a power series in this form, depending on the precise series.
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One possibility is that the series only converges when X equals A. A second possibility is that the series converges for any value of X.
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And lastly, there could be some positive number R such that if the absolute value of X minus A is less than R, the series will converge, and if greater than R, the series will diverge, just like the example we just completed.
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In this third case, that number R is called the radius of convergence for that power series.
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The first two cases also involve a radius of convergence. It is simply that for the first case, the radius of convergence is zero, as there is only one value that will allow for convergence, and for the second case, the radius of convergence is infinite, as
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any value will allow for convergence. We can also describe the interval of convergence as the interval describing all the values of X for which the series converges.
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Let’s try an example and see how this applies. Take X to the N, over N factorial.
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What can we do with this? Let’s try the ratio test. We get X to the N plus one over N plus one factorial, and then given the original series in the denominator, we just multiply by the reciprocal, or N factorial over X to the N.
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We’ve simplified something like this a couple times by now, so let’s change N plus one factorial into N plus one times N factorial, and let’s change X to the N plus one into X to the N times X to the one.
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Most of this cancels out, and we are left with X over N plus one.
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Finding the limit as N approaches infinity, we see that we will get zero no matter what the value for X.
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This means that the radius of convergence is infinity, and the interval of convergence is negative infinity to positive infinity.
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This fits the second possibility from that theorem, which tells us that this series is always convergent.
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That covers the basics regarding power series, so let’s check comprehension.
Topics:power seriesconvergencedivergenceratio testradius of convergenceinterval of convergencegeometric seriesinfinite seriescoefficientsmathematics
