Category Archives: Analysis

Taylor Series Uses #1: Complicated Integrals

1. Can you compute ? Well, first we consider the antiderivative for …which doesn’t exist. What do we do? Cry. No, what I mean is, use Taylor series! Advertisements

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Taylor Polynomials

1. So last time we concluded discussing Taylor series with constructing a polynomial using the first terms in the Taylor series. This “Taylor polynomial” approximated our function, and we want to know how well does it approximate? We will derive … Continue reading

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Taylor Series

0. Can we approximate a function using a power series? With the calculus’ help, we can!

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Power Series

1. Definitions. Let be a variable. a series of the form (1) is called a “Power Series about ”. A series of the form (2) is called a “Power Series about ”. Example 1. Consider the series (3) For what … Continue reading

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Other Alternating Series Tests

1. We have another couple of methods testing if an alternating series converges. It’s worth knowing as many different ways as possible, because sometimes one doesn’t work well (or at all). We will consider a couple tests. For each test … Continue reading

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Alternating Series Test

We have discussed whether a series will converge, when for each . But what about when we let be anything? What happens to ? When we expand it out, we see that it looks like . But by comparison for … Continue reading

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Comparison Tests

1. So if we have some complicated series, say (1) How can we tell if this behemoth converges or diverges? We can tell each term . So what? Well, we know by the integral test the series converges. Since each … Continue reading

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