File Name: sequences and series convergence tests .zip
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In the previous section we saw how to relate a series to an improper integral to determine the convergence of a series. Nicely enough for us there is another test that we can use on this series that will be much easier to use. As with the Integral Test that will be important in this section. Since all the terms are positive adding a new term will only make the number larger and so the sequence of partial sums must be an increasing sequence. Therefore, the sequence of partial sums is also a bounded sequence. Then from the second section on sequences we know that a monotonic and bounded sequence is also convergent.
The two simplest sequences to work with are arithmetic and geometric sequences. Sequences and Series. Determine the convergence or divergence of the sequence with the given nth term. Series With Negative Terms So far, almost all of our discussion of convergence and divergence has involved positive series. State the test used. Added Apr 17, by Poodiack in Mathematics.
In mathematics , a series is the sum of the terms of an infinite sequence of numbers. The n th partial sum S n is the sum of the first n terms of the sequence; that is,. Any series that is not convergent is said to be divergent or to diverge.
In a moment, you will open the packet that contains your exam materials. Problem 2 10 points Compute the exact value of. Exercise: Sketch the graph of the piecewise-defined functions x x2, if x 1 f x Math is the second semester of the standard three-semester calculus sequence. It continues the study of calculus on the real line, started in Math Calculus I , focusing on integration, the basics of sequences and series, and parametric descriptions for sets in the plane.
8.4: Convergence Tests - Comparison Test
We have seen that the integral test allows us to determine the convergence or divergence of a series by comparing it to a related improper integral. In this section, we show how to use comparison tests to determine the convergence or divergence of a series by comparing it to a series whose convergence or divergence is known. Typically these tests are used to determine convergence of series that are similar to geometric series or p-series. In the preceding two sections, we discussed two large classes of series: geometric series and p-series. We know exactly when these series converge and when they diverge.
Identify arithmetic and geometric series P. Find the sum of a finite arithmetic or geometric series P. Introduction to partial sums P.
Then the following rules are valid:. According to the Root Test:. Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.