Posts tagged series
Finding radius and interval of convergence of a Taylor series

Sometimes we’ll be asked for the radius and interval of convergence of a Taylor series. In order to find these things, we’ll first have to find a power series representation for the Taylor series.

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Using the comparison test to determine convergence or divergence

The comparison test for convergence lets us determine the convergence or divergence of the given series by comparing it to a similar, but simpler comparison series. We’re usually trying to find a comparison series that’s a geometric or p-series, since it’s very easy to determine the convergence of a geometric or p-series.

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The alternating series estimation theorem to estimate the value of the series and state the error

The alternating series estimation theorem gives us a way to approximate the sum of an alternating series with a remainder or error that we can calculate. To use the theorem, the alternating series must follow two rules.

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Sum of the maclaurin series

One convenient way to find the sum of the Maclaurin series is to start with a well-known Maclaurin series and then manipulate it one step at a time until it matches the series you’ve been given. Because you’ll be manipulating the expression of the sum at the same time, once you get the series to match, you’ll automatically have the sum.

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How to find the error or remainder of a series

If you need to find the sum of a series, but you don’t have a formula that you can use to do it, you can instead add the first several terms, and then use the integral test to estimate the very small remainder made up by the rest of the infinite series. The sum of the series is usually the sum of the first several terms, plus a very smaller error that you can estimate.

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