Practice exam question on series

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SUMMARY

The discussion centers on evaluating the infinite series Σ 2/((n+1)(n+3)) from n=1 to infinity. The integral test was initially applied, confirming convergence but not yielding the exact sum. The correct value of the series is established as 5/6. The confusion arises from the application of the Partial Sum Formula, which is not well-documented in the referenced materials.

PREREQUISITES
  • Understanding of infinite series and convergence tests
  • Familiarity with the Integral Test for convergence
  • Knowledge of the Partial Sum Formula: S_n = k/2 (a_1 + a_k)
  • Concept of telescoping series and their properties
NEXT STEPS
  • Study the application of the Partial Sum Formula in detail
  • Learn about telescoping series and how to derive their sums
  • Explore examples of the Integral Test for various series
  • Review convergence criteria for infinite series in calculus
USEFUL FOR

Students studying calculus, particularly those focusing on series convergence and summation techniques, as well as educators seeking to clarify concepts related to infinite series.

Matt Armstrong

Homework Statement


Find the value of the sum

(Infinity)
Σ 2/((n+1)(n+3))
(n=1)

Homework Equations



Integral test

Partial Sum Formula = k/2 (a_1 + a_k)

The Attempt at a Solution



Admittedly I started off this problem the wrong way. I used the integral test thinking I might get an answer there, but only found that it converged, not that it was equal to what I got. A little puzzled at the usage of 'partial sum formula' but I do not recognize it. Looking through my book's index, 'partial sum' isn't even mentioned. I flipped through the relevant chapters and didn't see mention of it either.

But, it still confuses me as to how it works. If I plug in an arbitrarily large number, like 1 billion, I get well above the correct answer, which was marked as 5/6. How do I properly apply the Partial Sum formula?
 
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Matt Armstrong said:

Homework Statement


Find the value of the sum

(Infinity)
Σ 2/((n+1)(n+3))
(n=1)

Homework Equations



Integral test

Partial Sum Formula = k/2 (a_1 + a_k)

The Attempt at a Solution



Admittedly I started off this problem the wrong way. I used the integral test thinking I might get an answer there, but only found that it converged, not that it was equal to what I got. A little puzzled at the usage of 'partial sum formula' but I do not recognize it. Looking through my book's index, 'partial sum' isn't even mentioned. I flipped through the relevant chapters and didn't see mention of it either.

But, it still confuses me as to how it works. If I plug in an arbitrarily large number, like 1 billion, I get well above the correct answer, which was marked as 5/6. How do I properly apply the Partial Sum formula?

Write it as a "telescoping series".
 

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