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/
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