In this section, we investigate convergence of series that are not made up of only non- negative terms.

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Presentation transcript:

In this section, we investigate convergence of series that are not made up of only non- negative terms.

Thus far, all of our tests for convergence are only for a series where a k ≥ 0 for all k. What about series where there are infinitely many k for which a k < 0?

If converges, then so does and Note: we can use all of our tests for convergence to determine whether or not converges since the terms are positive.

Determine whether or not the following series is absolutely convergent. If it converges, find N so that.

Just because a series does not converge absolutely, it does not necessarily follow that the series diverges. is a special type of series.

Let. If and Then the series converges and: 1. S is between any two consecutive partial sums. 2. For all n,

Determine whether the given series converges absolutely, converges conditionally, or diverges. If it converges, find upper and lower bounds for the sum.

Determine whether the given series converges absolutely, converges conditionally, or diverges. If it converges, find a partial sum that approximates S within 0.01.

Determine whether the given series converges absolutely, converges conditionally, or diverges. If it converges, find a partial sum that approximates S within 0.1.