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Summing Series (Sigma Notation)

  • #1

Homework Statement



a) nƩr2(r-1)r=1

Homework Equations



Using the summation series formulae...

The Attempt at a Solution



So far I have got:

r2(r-1) = r3-r2

Ʃr3 = [itex]\frac{1}{4}[/itex]n2 (n+1)2

Ʃr2 = [itex]\frac{1}{6}[/itex]n(n+1)(2n+1)

Therefore,

Ʃr3-r2 = [itex]\frac{1}{4}[/itex]n2 (n+1)2 - [itex]\frac{1}{6}[/itex]n(n+1)(2n+1)

But how do I subtract these two massive things from each other?!?!

Thanks.
 

Answers and Replies

  • #2
eumyang
Homework Helper
1,347
10
They aren't that massive. Rewrite them as fractions:
[tex]\frac{n^2(n+1)^2}{4}-\frac{n(n+1)(2n+1)}{6}[/tex]
Find a common denominator, rewrite both fractions so that both have that common denominator, and then subtract. It will be possible to factor things out.
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,808
933
It's pretty much just subtracting fractions. Get a common denominator:
The least common denominator of 4 and 6 is 12. To change the first denominator to 12, multiply both numerator and denominator by 3.
[tex]\frac{n^2(n+1)^2}{4}= \frac{3n^2(n+1)^2}{12}[/tex]

To change the second denominator to 12, multiply both numerator and denominator by 4.
[tex]\frac{4n(n+1)(2n+1)}{12}[/tex]

To subtract now, subtract the numerators:
[tex]\frac{3n^2(n+1)^2}{12}- \frac{4n(n+1)(2n+1)}{12}[/tex]

You will need to multiply those:
[itex]3n^2(n+1)^2= 3n^2(n^2+ 2n+ 1)= 3n^4+ 6n^3+ 3n^2[/itex]
[itex]4n(n+1)(2n+1)= 4n(2n^2+ 3n+ 1)= 8n^3+ 12n^2+ 4n[/itex]

and now subtract.
 
  • #4
They aren't that massive. Rewrite them as fractions:
[tex]\frac{n^2(n+1)^2}{4}-\frac{n(n+1)(2n+1)}{6}[/tex]
Find a common denominator, rewrite both fractions so that both have that common denominator, and then subtract. It will be possible to factor things out.
Yeah ok, I've done that, but the textbook answers want it to be "factorised", and I'm unsure how to do that after subtracting... Thanks for your help.
 
  • #5
@HallsOfIvy

Hmm thanks, for the help, I'll try to factorise those now, since the book wants it to be factorised.
 
  • #6
HallsofIvy
Science Advisor
Homework Helper
41,808
933
You will have a quadratic factor that cannot be factored further (with integer coefficients).
 

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