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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.

Artem


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

  http://mathworld.wolfram.com/BernoulliNumber.html   

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 
number)." 

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=1,n=m,(n+1)^2) 
  = 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.

  http://mathforum.org/dr.math/problems/kijjaz11.24.98.html   

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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