Maths

I'm having problems with a question, just don't know how to approach. So I'm given an infinite series and I'm asked to write an expression for it in terms of n. Here's the series:
n
Σ 3r+4/r(r+1)(r+2)
r=1

n on top and r on bottom of the sigma sign.

Any help would be appreciated.

Also, the first part is show that 2/r - 1/r+1 -1/r+2 = the sum I gave above

Are you able to do the first part? It splits the series into three separate series each of which are easier to compute

Yes first part is fine but don't know how to find the expression in terms of n

Are you familiar with the method of differences?

Yes but I'm not sure how to apply it in this case.

You need to sub in a value of r three times for each value, once for each of 2/r, -1/r+1, -1/r+2

Could you show me how it's done? Nothing seems to cancel out here for me...

r=1: 2 -1/2 -1/3

r=2: 1 -1/3 -1/4

r=3: 2/3 -1/4 -1/5

r=4: 1/2 -1/5 -1/6

See how the 2/3 in line 3 cancels out the -1/3 in lines 1 and 2? Same for the 1/2 in line 4 cancelling out the -1/4s?

Oh ok Yes. What happens after this since we still have a few values left?

Just like normal, you add up the values left at the start and the values made of 'n's at the end

Is the values of n what u get when u sub in r=n?

Since there are three values left over at the start, there will also be three left over at the end. You have to look at the shape of the cancellations to know which ones, but one of them will be from r = n-1

Thanks so much! I got it, you think you can help me with the next part, it’s the same series but the limit (on top of sigma) is infinity. How would I go about finding the sum of that??

My initial instinct is that n being infinity would mean that every value after the first three values in the method of differences table would cancel out, because there would always be another line to cancel the previous lines

Yep that makes sense when u put it like that, so it should be 2.5?

I think so! Haven't done this stuff for a while but that approach makes sense to me