Write it as a partial fraction. ?/(n+1) + ??/(n+2) + ???/(n+3). Then you see that it is a telescoping series.
Isabella M.
asked • 03/20/19Series n/((n+1)(n+2)(n+3)) from 1 to infinity is equal to 1/4. Why?
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2 Answers By Expert Tutors
John T. answered • 03/20/19
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First, the series converges as demonstrated by either the direct comparison test or the limit comparison test. You can compare this series to 1/n2, a convergent p-series. With these tests, if the limits of the two expressions do the same thing, then the two series both converge or both diverge. Also, you should show that as n increases the series of interest produces smaller elements than does 1/n2.
Next, if you program Excel to find the first 400 terms and the sum of the first 400 terms, you'll see the sum, rounded to three decimal places, is 0.248. Moving to the 0.249 is going to take many more terms. You can approximate by finding the integral of the sequence explicit form with respect to n from 0 to 400 and have a fair estimate of 0.259.
Steven G.
tutor
Doctoral level math tutor using excel?? No comment at all
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03/20/19
Jason P.
Steven, thanks for the comment. Please be sure you treat other users on AAE respectfully, and that when you reply, you are providing help for a question.
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03/21/19
John T.
tutor
Steven, my graduate training is in physiology, cell biology, and public health. Sometimes students can see things from the non-subject instructor perspective more clearly, just like I presume you might when, say, examining the mechanisms of alkane bromination via radical intermediates. The reason for Excel is because the series was not approaching 1/4 very quickly. It was nothing more than a convenience. Your explanation that the series is telescoping after written as a partial fraction is more concise. Thank you and have a great day.
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03/21/19
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Isabella M.
I did that, and I got -1/2(n+1) + 2/(n+2) - 3/2(n+3), but I don't see the form of a telescoping series.03/20/19