The 'upper bound' parameter in capital-sigma notation
Favorites|Homepage
Subscriptions | sitemap
HOME > > The 'upper bound' parameter in capital-sigma notation

The 'upper bound' parameter in capital-sigma notation

[From: ] [author: ] [Date: 12-06-21] [Hit: ]
where the index is k, the lower bound is k = 1, and the upper bound is 2n.S = 2(1^2 + 2^2 + .........
I'm teaching myself pre-calculus mathematics. There are plenty of accessible resources on the net showing what the parameters of capital-sigma notation do, but I'm finding there's not enough variation in their examples for me to deduce all the facts...

Let's say there's an arithmetic sequence S of k^2, where the index is k, the lower bound is k = 1, and the upper bound is 2n.

Does that mean we have
S = 2(1^2 + 2^2 + ... + n^2),
or does it mean
S = 1^2 + 2^2 + ... + (2n)^2?

Or does it mean something else?

Also, is my wording correct? I'm particularly unsure of the portion, "there's an arithmetic sequence S of k^2".

-
This is a good question!

That particular sigma notation means the second interpretation you have listed, namely

1^2 + 2^2 + ... + (2n)^2

(a sum with exactly 2n terms in it).

I would read it aloud as "the sum, from k equals one to two n, of k squared" or "the sum of k^2, from k equals one to two n."

If you wanted to indicate 2 (1^2 + 2^2 + ... + n^2), two equivalent ways of doing this would be to write it as

[capital sigma] [k = 1 on the bottom of the sigma] [n on the top of the sigma] 2k^2,

or

2 [capital sigma] [k = 1 on the bottom of the sigma] [n on the top of the sigma] k^2.

This sum would generally not be called the sum of an "arithmetic" sequence, or an "arithmetic series", because successive terms in the sum do not differ by a constant amount. Arithmetic sequences are things like 1, 2, 3, 4, 5 (successive terms differ by the constant 1), or 1, 3, 5, 7, 9 (successive terms differ by the constant 2) or 3, 6, 9, 12, ... (successive terms differ by the constant 3) and so on.

I hope this helped! Sigma notation is a great tool, once you get used to it. It takes a bit of time to learn, but once you get acquainted with it, you can manipulate sums with variable numbers of terms algebraically, without thinking--- sort of like how learning basic algebra lets you manipulate formulas with a fixed number of terms algebraically, without thinking.
1
keywords: parameter,upper,notation,capital,The,bound,in,039,sigma,The 'upper bound' parameter in capital-sigma notation
New
Hot
© 2008-2010 http://www.science-mathematics.com . Program by zplan cms. Theme by wukong .