Pointwise convergence of (x^n)/(1 + x^n)

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In summary, pointwise convergence is when a sequence of functions converges to a specific value at a particular point. For the function (x^n)/(1 + x^n), pointwise convergence is determined by the limit of the function at each point, which is 0. Therefore, the function is pointwise convergent for all real numbers. Pointwise convergence is different from uniform convergence, as the former is a property at a specific point while the latter means convergence at every point. Pointwise convergence is also necessary for the convergence of the corresponding series of functions.
  • #1
youko.shi
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Homework Statement


Find the function that (x^n)/(1 + x^n) converges to as n goes to infinity, on the interval [0,2]


Homework Equations





The Attempt at a Solution


I've worked out the fact that on the interval [0,1) it converges to 0, and when x is 1 it converges to 1/2, but for the life of me I'm not sure what the first step of proving what it converges to on the (1,2] interval. hints?
 
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  • #2
Divide numerator and denominator by x^n. Now can you figure out what it converges to on (1,2]?
 
  • #3
Yes! It becomes 1 / ( 1/(x^n) + 1), which goes to 1 / (0+1) = 1.

Thank you so much :)
 

FAQ: Pointwise convergence of (x^n)/(1 + x^n)

1. What is pointwise convergence?

Pointwise convergence refers to the behavior of a sequence of functions at a particular point. It means that as the independent variable (x) goes to a specific value, the values of the function also converge to a specific value.

2. How is pointwise convergence determined for the function (x^n)/(1 + x^n)?

For a sequence of functions to be pointwise convergent, the limit of each function at a given point must exist and be equal to the limit of the sequence at that point. In the case of (x^n)/(1 + x^n), the limit at any point x is 0, which means the sequence of functions is pointwise convergent.

3. What is the range of values for which (x^n)/(1 + x^n) is pointwise convergent?

Since the limit of the function (x^n)/(1 + x^n) is 0 for all values of x, the function is pointwise convergent for all real numbers.

4. Is pointwise convergence the same as uniform convergence?

No, they are not the same. Pointwise convergence is a property of a sequence of functions at a particular point, while uniform convergence means that the functions converge to a specific value at every point in their domain.

5. How is pointwise convergence related to the convergence of a series?

Pointwise convergence of a sequence of functions is necessary for the convergence of the corresponding series of functions. If the sequence of functions is not pointwise convergent, the series of functions will not converge either.

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