Summation Notation and Arithmetic Series

Date: 07/27/2001 at 16:58:45
From: rachel
Subject: Summation Notation and Arithmetic Series

I have to use sigma notation with arithmetic series. I know how to use 
the equations for arithmetic series, but when I go to use sigma 
notation it seems as though 
a. I don't need these formulas, or 
b. I don't fully understand how to use k= and plug it in.

Ex.  3+(4-1)3   3 = a   4 = n   3 = d

What is k when I go to put it into sigma? Is it 3, or is it 1? 3 would 
mean it was f(k), and 1 would mean that I would need to multiply by 
d (3).

Or do I not need to use the arithmetic series formulas when doing 
sigma notation, for I could just have sigma with k = 1, n = 4, 
followed by 3k.

Thank you.

Date: 07/27/2001 at 23:09:19
From: Doctor Peterson
Subject: Re: Summation Notation and Arithmetic Series

Hi, Rachel.

The sigma notation really is just a different way to write the series 
itself, and has nothing to do with finding the sum. There, you still 
use the same formula when the time comes to find the sum.

You haven't actually given me a series, and I'm not quite sure what it 
is that you did show. I suppose a is the first term, n is the number 
of terms, and d is the common difference, so your series is

    3 + 6 + 9 + 12

with four terms, starting at 3 and increasing by 3 each time. Your 
3+(4-1)3 must be the formula for the last term; the general kth term 
would be

    a_k = 3 + (k-1)*3

so that, for instance, when k = 1 it gives 3+(1-1)*3 = 3, and when 
k = 2 it gives 3+(2-1)*3 = 6. For each term in the series, k has a 
different value, 1, 2, 3, and 4. So k is not a specific number, but 
the index of any term. That is, it's a variable that changes from one 
term to another.

The series, then, is the sum of these terms, with k taking values from 
1 through 4. That's expressed in sigma notation this way:

      4
    +---
     \
     /   3 + (k-1)*3
    +---
     k=1

In this, I've said nothing I haven't said hefore: you can see the 
formula for the kth term, and the range of values taken by k, from 1 
to 4. So this merely shows in an orderly way what the series is; it is 
just an alternative to writing out every term and saying

    3 + 6 + 9 + 12

Now, if you are given the series in this form, and you want to find 
the sum, you have to do the same thing you'd do if you were given it 
in any other form: determine the values of a, d, and n (and the fact 
that it is, indeed, an arithmetic series), and plug them into the 
formula for the sum.

Yor very last question suggests that you saw the right answer. As you 
pointed out, our formula for the kth term can be simplified, so that 
the series becomes

      4
    +---
     \
     /   3k
    +---
     k=1

This makes it a little less obvious, perhaps, what the parameters of 
the series are. You can see from the fact that the kth term is linear 
that this is an arithmetic series, and by setting k = 1 can find the 
first term.

If this doesn't fully answer your question, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/