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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!
Posted in Analysis, Calculus, Infinite Series, Power Series, Taylor Series
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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
Posted in Analysis, Calculus, Differential, Infinite Series, Power Series, Taylor Series
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Taylor Series
0. Can we approximate a function using a power series? With the calculus’ help, we can!
Posted in Analysis, Approximation, Calculus, Differential, Infinite Series, Power Series, Taylor Series
Tagged Power Series, Taylor Series
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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
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
Posted in Analysis, Calculus, Infinite Series
Tagged Alternating Series, Convergence Tests
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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
Posted in Analysis, Calculus, Infinite Series
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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
Posted in Analysis, Calculus, Infinite Series
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Integral Test for Convergence
1. So we have the definition for a series to converge, but what about some slick tests? Wait a moment: wasn’t a Riemann Sum an infinite series? Why don’t we use integration to test if a series converges or diverges? … Continue reading
Posted in Analysis, Approximation, Calculus, Infinite Series
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Series Convergence: What Does It Mean?
1. So we discussed what it means for a sequence to converge, and we are really interested in infinite series. It makes sense to first ask “Does this series equal a finite number?” BEFORE we ask “What number is this … Continue reading
Posted in Analysis, Calculus, Infinite Series
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L’Hopital’s Rule for Limits
1. Now we are concerned about limits and we can have nightmarish limits. For example if (1a) then (1b)? What can we do? We end up using derivatives to help evaluate such limits. We have L’Hopital’s rule state, if , … Continue reading
Posted in Calculus, Derivative, Limits, Sequences
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