On infinite products; notational questions

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In summary: I hope it helps!In summary, the conversation discusses the concept of "big O" in mathematics and how it is used to represent the rate of growth of a function in relation to another function. This is demonstrated through examples and explanations of how to use "big O" notation. The conversation also delves into the use of Latex for mathematical formulas and the posting of pictures as a means of displaying equations.
  • #1
DoubleRaven
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I'm not certain this is the right forum, so let me know if its not, but I was wondering:

I'm reading a book on special functions and there are infinite products involved. In showing they converge, a capital O crops up and is not defined. Can anyone help me? I don't know how to paste in math formulas using latex (something I would like to learn if anyone can help me with that), otherwise I would. As is I have posted a picture I did with MathType.

What is the O mean? Perhaps that answer will be enough for me to answer the next on my own, but in case not: how do they arive at the second line's equality?

Thanks for any help you can offer.
 

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O in this context means Order. Since it's a "Big O" (as opposed to "little o"), it represents that the cropped terms are at most in the order of -3 (i.e. somewhat small). Formally, f(n) = O(g(n)) means that "the rate of growth of f(n) is no more than a constant times the rate of growth of g(n)." See http://mathforum.org/library/drmath/view/54574.html. In your case, n = -3.
 
  • #3
Enuma has as such detailed out what big-Oh denotes.

To top it off with a small example,
consider this polynomial,
f(n) = 1+n+n^2+n^3+n^4 (for all n>=1)

Now if i can find two constants n_0 and c, such that
f(n) <= c*g(n) for all n>=n_0
then we say that
f(n) = O(g(n))
(**This is what Enuma states as, "the rate of growth of f(n) is no more than a constant times the rate of growth of g(n)." **)

Consider g(n) = n^4,
and now consider the constants n_0 = 1 and c = 5, you will notice that,
f(n) <= 5*g(n) for all n>=1 (Try to prove this if you wish)
hence f(n) = O(g(n))
or (1+n+n^2+n^3+n^4) = O(n^4)

Now moving a bit further, consider this polynomial,
f(n) = n^5 + (1+n+n^2+n^3+n^4) (for all n>=1)
I can always replace the term in bracket with,
f(n) = n^5 + O(n^4)

Such a replacement is done when we are only concerned with the growth rate of the term and not the actual term itself.

-- AI
 

What is an infinite product?

An infinite product is a mathematical concept that represents the multiplication of an infinite sequence of numbers. It is denoted by the symbol Π (capital pi), and is similar to an infinite sum, but instead of addition, multiplication is used.

How is an infinite product notated?

An infinite product is typically notated using the symbol Π followed by the terms being multiplied. For example, Πn=1∞ an represents the product of all terms an from n=1 to infinity.

What is the difference between an infinite product and a finite product?

A finite product is the multiplication of a finite sequence of numbers, while an infinite product involves multiplying an infinite sequence of numbers. In a finite product, the number of terms is known and finite, while in an infinite product the number of terms is infinite and can continue indefinitely.

What is the convergence of an infinite product?

The convergence of an infinite product refers to whether the product of its terms approaches a finite nonzero value as the number of terms increases. If the product approaches a finite nonzero value, the infinite product is said to converge. If the product approaches infinity or zero, the infinite product is said to diverge.

How is the value of an infinite product determined?

The value of an infinite product can be determined by evaluating the limit of the product as the number of terms approaches infinity. Various techniques, such as the Ratio Test and the Root Test, can be used to determine the convergence or divergence of an infinite product.

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