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Sigma Notation

Date: 12/17/98 at 01:11:15
From: Artem Pyatakov
Subject: Calculus

Dear Dr. Math,

We just learned about sigma notation and summation formulas in my 
calculus class. Unfortunately, our teacher only showed us 4 formulas: 
("Sum" refers to the symbol sigma)

    Sum(k), Sum(i), Sum(i^2), Sum(i^3)

We also learned that you can Sum(U1)+Sum(U2)=Sum(U1+U2) and that 

    Sum(ki) = kSum(i). 

However, I would like to know more on the sigma topic. Can someone 
please tell me more formulas? For example: what is the formula for 
Sum(i^k) (i raised to any power)? Also what is the formula for sum(i), 
but not starting with 1 (any number). 

Do you know of an easy way to find Sum((n+1)^2), I mean without 
distributing and then doing separate sums (I have been experimenting 
with changing bounds but it's hard without other formulas).

I am getting no sleep trying to figure out some formulas, so please 
help me.


Date: 12/17/98 at 08:09:29
From: Doctor Jerry
Subject: Re: Calculus

Hi Artem,

"What is the formula for Sum(i^k) (i raised to any power)?" 

There is a formula, one discovered a long time ago by Bernoulli.  See 
the web site   

You will have to look through several pages of stuff until you come to 
what you want. If the page is not available on the day you try, wait a 
few days and try again.

"Also what is the formula for sum >(i), but not starting with 1 (any 

I suppose you could write this as sum(k=m,k=n,k). I'm sure you can 
see that

   sum(k=1,k=m-1,k) + sum(k=m,k=n,k) = sum(k=1,k=n,k).

There is a formula for the first and last of these.

"Do you know of an easy way to find Sum((n+1)^2)"

Yes, just change the index of summation and then adjust a little.

Letting n+1=N,

  = sum(N=2,N=m+1,N^2) 
  = sum(N=1,N=m+1,N^2) - 1

then apply the standard formula for Sum(n^2).

You might also find the following link in our archive to be helpful.   

- Doctor Jerry, The Math Forum   
Associated Topics:
High School Number Theory

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