Newton's Generalized Binomial Theorem

In summary, the conversation discusses the use of Newton's Generalized Binomial Theorem to expand the expression $[f_1(x)+f_2(x)]^\delta$ under the condition that $0<\delta<<1$ and $\lvert f_1(x)\rvert > \lvert f_2(x)\rvert$. However, there are instances where the condition is violated and the expansion is not true for all values of x. The solution proposed is to use two cases, one for $\lvert f_1(x)\rvert > \lvert f_2(x)\rvert$ and another for $\lvert f_2(x)\rvert > \lvert f_1(x)\r
  • #1
JBD
15
1
I'm trying to expand the following using Newton's Generalized Binomial Theorem.
$$[f_1(x)+f_2(x)]^\delta = (f_1(x))^\delta + \delta (f_1(x))^{\delta-1}f_2(x) + \frac{\delta(\delta-1)}{2!}(f_1(x))^{\delta-2}(f_2(x))^2 + ...$$
where $$0<\delta<<1$$

But the condition for this formula is that $$\lvert f_1(x)\rvert > \lvert f_2(x)\rvert$$

And that's where my problem is. Since both functions are sinusoidal, there are times when indeed $$\lvert f_1(x)\rvert > \lvert f_2(x)\rvert$$ but there are also values of x such that $$\lvert f_2(x)\rvert > \lvert f_1(x)\rvert$$. Take for example the graphs of cos^2 x and sin^2x.

In other words, since the condition is violated, the expansion is not true for all x.

I'm thinking of separating the two instances. At x's where $$\lvert f_1(x)\rvert > \lvert f_2(x)\rvert$$ then I can use the above expansion. If $$\lvert f_2(x)\rvert > \lvert f_1(x)\rvert$$, then:

$$[f_2(x)+f_1(x)]^\delta = (f_2(x))^\delta + \delta (f_2(x))^{\delta-1}f_1(x) + \frac{\delta(\delta-1)}{2!}(f_2(x))^{\delta-2}(f_1(x))^2 + ...$$

But, how can I separate the two instances? Or is there another way to solve this problem?
 
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  • #2
You need the order for convergence of the series, and I don't see a way to avoid using two cases.
 
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  • #3
mfb said:
You need the order for convergence of the series, and I don't see a way to avoid using two cases.
How do you find the order of convergence? Do you mean use convergence tests? Most of the time f1 > f2 so I think the series converges. Does this mean that I'm allowed to use my first expansion? Thanks.
 
  • #4
It is a power series in ##\frac{f_2}{f_1}##, this fraction has to be smaller than 1 to make the power series converge. There is nothing to test.
If it is larger than 1, you can swap the two values as you did in post 1 and then use the formula with swapped values.
 
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Related to Newton's Generalized Binomial Theorem

1. What is Newton's Generalized Binomial Theorem?

Newton's Generalized Binomial Theorem is a mathematical formula that expands the power of a binomial expression (a + b)^n, where n is any real number, not just a positive integer. It is an extension of the traditional binomial theorem discovered by Sir Isaac Newton.

2. How is Newton's Generalized Binomial Theorem different from the traditional binomial theorem?

The traditional binomial theorem only works for positive integer exponents, while Newton's Generalized Binomial Theorem works for any real number exponent. This allows for a wider range of applications in mathematics and physics.

3. What is the formula for Newton's Generalized Binomial Theorem?

The formula is (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where n is the exponent, k is the term number, and (n choose k) is the binomial coefficient, also known as the combination formula.

4. How is Newton's Generalized Binomial Theorem used in science?

Newton's Generalized Binomial Theorem is used in various fields of science, including physics, chemistry, and biology, to expand and simplify complex mathematical expressions. It is particularly useful in studying the behavior of physical systems involving continuous variables.

5. Can Newton's Generalized Binomial Theorem be applied to any type of expression?

No, Newton's Generalized Binomial Theorem can only be applied to binomial expressions, which have two terms. It cannot be used for expressions with more than two terms, such as trinomials or polynomials.

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