Proving Series of Functions on (-1,1)

In summary, the task is to prove that the series \sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{2n+1} is well-defined and differentiable on (-1,1). This can be done by using the Uniform Cauchy Criterion to show that it converges uniformly, and then showing that it is the series expansion for arctan(x) by equating their derivatives. The maximum difference between terms can be found by choosing values for n, p, and x, and noting that the series is alternating.
  • #1
Whistlekins
21
0

Homework Statement



Prove that the series [itex]\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{2n+1}[/itex] is well-defined and differentiable on (-1,1).

Homework Equations





The Attempt at a Solution



I know that the function is the series expansion of arctan(x), but that it not we are showing here (however it asks in a later question to show that it is). I don't know what it means by "well-defined", but I'm going to guess it means continuous and convergent on its domain. I am guessing that I should use the Uniform Cauchy Criterion to show that it converges uniformly, and thus showing that it is differentiable.

But I'm not sure how to show that for all ε > 0, there exists and N ≥ 1 such that, for all n, p ≥ N and all x in (-1,1), |f_n(x) - f_p(x)| < ε

Also, how would I show that it is actually the series expansion for arctan(x)?

A nudge in the right direction would be greatly appreciated.
 
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  • #2
Whistlekins said:
I am guessing that I should use the Uniform Cauchy Criterion to show that it converges uniformly, and thus showing that it is differentiable.

But I'm not sure how to show that for all ε > 0, there exists and N ≥ 1 such that, for all n, p ≥ N and all x in (-1,1), |f_n(x) - f_p(x)| < ε

That sounds like a good plan. Can you see when |f_n(x) - f_p(x)| would be maximized? You can bound this from above by choosing a value for x that gives the largest difference.

Whistlekins said:
Also, how would I show that it is actually the series expansion for arctan(x)?

You can probably show that their derivatives are equal.
 
  • #3
clamtrox said:
That sounds like a good plan. Can you see when |f_n(x) - f_p(x)| would be maximized? You can bound this from above by choosing a value for x that gives the largest difference.

Can you expand on what you mean by maximized? Would that happen when n = 0 and p -> ∞?
 
  • #4
Whistlekins said:
Can you expand on what you mean by maximized? Would that happen when n = 0 and p -> ∞?

I mean choose value for n, then pick a p that maximizes the difference for arbitrary x and then choose x to find the absolute upper bound for a given n. And notice that the series is alternating, so n=0 and p→∞ is not the maximum.
 

What is meant by a series of functions on (-1,1)?

A series of functions on (-1,1) refers to a collection of functions that are defined on the interval from -1 to 1, which can be combined to form a single function. This allows for a more efficient and concise representation of a function, as well as the ability to analyze its behavior more easily.

How do you prove a series of functions on (-1,1)?

To prove a series of functions on (-1,1), you must first show that the series of functions converges on the interval (-1,1). This can be done using various convergence tests, such as the ratio test or the comparison test. Once the convergence is established, you can then show that the resulting function is continuous on (-1,1) and satisfies the given conditions.

What are some common techniques used to prove a series of functions on (-1,1)?

Some common techniques used to prove a series of functions on (-1,1) include using convergence tests, such as the ratio test or the comparison test, to show that the series converges. Additionally, you can use techniques such as differentiation and integration to show that the resulting function is continuous and satisfies the given conditions.

Why is proving a series of functions on (-1,1) important?

Proving a series of functions on (-1,1) is important because it allows for a more efficient and concise representation of a function, which can make it easier to analyze and understand its behavior. Additionally, proving a series of functions can also help to establish the convergence of a function, which is crucial in many mathematical and scientific applications.

What are some real-world applications of proving series of functions on (-1,1)?

Proving series of functions on (-1,1) has many real-world applications, including in engineering, physics, and economics. For example, in engineering, series of functions are often used to model physical systems, and proving these series allows for more accurate predictions and analysis. In economics, series of functions can be used to model economic trends and make forecasts, and proving these series can help to validate the models and their predictions.

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