# Thread: Double Sigma Notation Help

1. ## Double Sigma Notation Help

Hi guys,

I am having a little trouble understanding a multi sigma notation. I have a double sigma notation, the first sigma is directly followed by a second sigma with no arithmetic operator between them. i am given:

S(n) = (First Sigma) (Second Sigma) i

[the "i" is the formula after the second sigma. indexes : first sigma==>lower index :: k=1, upper index=n
second sigma==> lower index:: i=1, upper index=k]

i am asked to prove that

$S(n) =[ n(n + 1)(n + 2)]/6$

(I don't know how to make the sigma signs appear in this forum.)

I am asking if someone can explain to me what the double sigma notation means.

My understanding is this:
I replace the second sigma with a formula, so that the formula preceded by the first sigma is used for the proof .

Is that right?

Need help.

Thanks.

2. Originally Posted by kap
Hi guys,

I am having a little trouble understanding a multi sigma notation. I have a double sigma notation, the first sigma is directly followed by a second sigma with no arithmetic operator between them. i am given:

S(n) = (First Sigma) (Second Sigma) i

[the "i" is the formula after the second sigma. indexes : first sigma==>lower index :: k=1, upper index=n
second sigma==> lower index:: i=1, upper index=k]

So this seems to be $S(n)=\sum^n_{k=1}\sum^k_{i=1}i$ $=1+(1+2)+(1+2+3)+\ldots +(1+2+\ldots+ n)$.

Now, use that for any $k\in\mathbb{N}\,,\,\,1+2+\ldots +k=\frac{k(k+1)}{2}$ and $1^2+2^2+\ldots +k^2=\frac{n(n+1)(2n+1)}{6}$ (you can easily prove both formulae by

induction), and $S(n)=\sum^n_{k=1}\frac{k(k+1)}{2}=\frac{1}{2}\left (\sum^n_{k=1}k^2+\sum^n_{k=1}k\right)$ , plus a little algebra to deduce the result they want.

The double notation may be understood as follows: make the first index run, and for each value it takes make the second sum's index run.

Tonio

i am asked to prove that

$S(n) =[ n(n + 1)(n + 2)]/6$

(I don't know how to make the sigma signs appear in this forum.)

I am asking if someone can explain to me what the double sigma notation means.

My understanding is this:
I replace the second sigma with a formula, so that the formula preceded by the first sigma is used for the proof .

Is that right?

Need help.

Thanks.
.