Question

Find the sum:

Answer 1

\[\sum_{k=5}^{\infty}\frac{ 4 }{ k^2-k }\]

Answer 2

I though this would be a geometric series, but i dont think it is, and I cant seem to apply the basic divergence test either.

Answer 3

\[\frac{ 4 }{ k ^{2} - k } = \frac{ 4 }{ k(k-1) }\]
Try using partial fractions.

Answer 4

\[\frac{ 4 }{ k-1 } - \frac{ 4 }{ k }\]
So the answer you need is:
\[\frac{ 4 }{ 4 } - \frac{ 4 }{ 5 } + \frac{ 4 }{ 5 } - \frac{ 4 }{ 6 } + ......\]

Answer 5

As you can see, many terms will get cancelled out, all that remains will be:
\[1 - \frac{ 4 }{ \infty + 1 } = 1 - 0 = 1\]

Answer 6

Where do those negative terms come from? Why doesn't it gust go
4/4 + 4/5 + 4/6 + 4/7 etc ?

Answer 7

@wolf1728 those negative terms come out as a result of expressing the components of the sum as partial fractions as Prathamesh has shown.

Answer 8

Ah I see. Im not familiar with partial sums yet, which explains why I didnt know how to do it. Im going to watch a few youtube videos on it and try it again