Which series will converge

Example of Divergent Series. Extension: Finite Telescoping Sums and Products. Infinite Geometric Series. Interpreting Function Notation, Graphs, and Context.

View All Related Lessons. You've reached the end. Here is a nice set of facts that govern this idea of when a rearrangement will lead to a different value of a series.

This is here just to make sure that you understand that we have to be very careful in thinking of an infinite series as an infinite sum. There are times when we can i. Eventually it will be very simple to show that this series is conditionally convergent. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i.

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Example 1 Determine if the following series is convergent or divergent. If it converges determine its value. Example 2 Determine if the following series converges or diverges. If it converges determine its sum. Example 3 Determine if the following series converges or diverges.

Example 4 Determine if the following series converges or diverges. Example 5 Determine if the following series is convergent or divergent. So I'll graph this as our y-axis. And I'm going to graph y is equal to a sub n. And let's make this our horizontal axis where I'm going to plot our n's. So this right over here is our n's. And this is, let's say this right over here is positive 1.

This right over here is negative 1. And I'm not drawing the vertical and horizontal axes at the same scale, just so that we can kind of visualize this properly. But let's say this is 1, 2, 3, 4, 5, and I could keep going on and on and on. So we see here that when n is equal to 1, a sub n is equal to 1. So this is right over there. So when n is equal to 1, a sub n is equal to 1. So this is y is equal to a sub n. And we keep going on and on and on. So you see the points, they kind of jump around, but they seem to be getting closer and closer and closer to 0.

Which would make us ask a very natural question-- what happens to a sub n as n approaches infinity? If and are convergent series , then and are convergent.

If , then and both converge or both diverge. Convergence and divergence are unaffected by deleting a finite number of terms from the beginning of a series. Constant terms in the denominator of a sequence can usually be deleted without affecting convergence. All but the highest power terms in polynomials can usually be deleted in both numerator and denominator of a series without affecting convergence. If the series formed by taking the absolute values of its terms converges in which case it is said to be absolutely convergent , then the original series converges.

Conditions for convergence of a series can be determined in the Wolfram Language using SumConvergence [ a , n ]. In contrast, the sums. Baxley ; Braden ; Zwillinger , p. OEIS A ; Mathar converge by the integral test , although the latter converges so slowly that terms are needed to obtain two-digit accuracy Zwillinger , p. Both can be summed using the Euler-Maclaurin integration formulas.