Practice exam question on 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?

 

[Ray Vickson]

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".