Finding the Sum of n/[(n+1)(n+2)(n+3)]

[Feynmanfan]

Could anybody please tell me how to do this?

Σn/[(n+1)(n+2)(n+3)]

Thanks!

 

[Zurtex]

You tried splitting up into partial fractions and using the method where you have one function minus another?

 

[Feynmanfan]

I've already splitted it. It's interesting that each fraction alone diverges but the whole sum converges. But now I don't know what to do. What is this method you told me about?

Thanks

 

[MathematicalPhysicist]

as zurtx said you can split it like this : Σ(1/[n+3])*(1/[n+1]-1/[n+2])=1/[n+3]*Σ1/[n+1]-1/[n+2]).

 

[Feynmanfan]

Still can't solve it. My computer says it certainly converges to 1/4. How can I prove that?

 

[MathematicalPhysicist]

Still can't solve it. My computer says it certainly converges to 1/4. How can I prove that?

did you learned the term when a series is telescoped (or something of that that sort i have it on a book in hebrew in level of calclus 2)?

 

[matt grime]

LQG, you can't take that term outside the summation as the n is part of the summation index.

 

[MathematicalPhysicist]

<acronym title='Loop Quantum Gravity' style='cursor:help;'>LGQ</acronym>, you can't take that term outside the summation as the n is part of the summation index.

it cannot be representing a coefficient in the form of Σd*(a1-an)
where d=1/(n+3) is the coefficient?

 

[matt grime]

Apologies for getting LGQ and <acronym title='Loop Quantum Gravity' style='cursor:help;'>LGQ</acronym> mixed up (if it's any consolation I'm reading a paper on TQFT and such as I write (well, not exactly "as I write" but you get the idea))

You're taking the sum indexed by n when you say 1/[n+3]*Σ1/[n+1]-1/[n+2] ie n runs from something to something. You can't take the n+3 outside like that.

It's just like taking the x outside an integral wrt x (of course it's possible that it is the interior part of the summation that is wrong)

 

[Feynmanfan]

matt grime is right. Yet that doesn't solve the problem. Thanks anyway for your help

 

[matt grime]

but you can do it by separating into partial fractions, it's quite easy isn't it?

 

[Zurtex]

Where A, B, C are arbitrary constants let:

\frac{n}{(n+1)(n+2)(n+3)} = \frac{A}{n+1} + \frac{B}{n+2} + \frac{C}{n+3}

Work out A, B and C. Now I could be wrong but I got A and C as negative numbers. So you should be able to write out the functions first few values n= 1, 2, 3, 4 and spot a pattern that if you add them together terms will start to cancel and you can extend that to n = r-3, r-2, r-1, r and get a general equation for the sum of f(n) from n = 1 to r.

 

[Anteros]

I just worked this out and the answer is in fact 1/4. Zurtex has the right approach. Find A, B and C and then write out the first several terms for each partial fraction. You should see a bunch of them cancel and you'll be left with a couple terms per fraction. From there, you can see the limit.

 

[Feynmanfan]

Thanks everybody! You're all great.