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Unformatted text preview: est, we consider the limit of the ratio an+1/an. The ratio test is inconclusive for this series.
9. Use the alternating series test to show that the series
converges. Then estimate the error
made by using the partial sum S9 as an approximation to the sum S of the series.
First, this is an alternating series, at least for n ≥ 3.
Second, each term
is positive, at least for n ≥ 3.
Third, since the function has the derivative
2 which is negative for x ≥ e , then this function is decreasing on [e , ∞). So
sufficiently large n.
Fourth, but not least, for all
. So the series meets the conditions of the alternating series test. And, by that test, the series converges.
The error made by using the partial sum S9 as an approximation to the sum S of the series is bounded by the
. Part III: Power Series.
10. Find the convergence set for the series
I'll begin with the absolute ratio test. .
. This converges for all x-values for which |x – 2| < 1. More simply, for 1 < x < 3. To finish up, I must
check for convergence at the endpoints of this interval.
For x = 1, we have the series . This is the harmonic series, which diverges. And...
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This document was uploaded on 03/18/2014 for the course MATH 237 at Frostburg.
- Fall '1