The title could be: Uniform Convergence of f_n(x) = nx^n(1-x) on [0,1]?

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In summary, the limit of f_n(x) is 0 and f_n converges uniformly on [0,1] because the supremum of |f_n(x) - f(x)| is equal to n(1/2)^n(1-1/2) and the limit as n approaches infinity of the supremum is 0. Additionally, by choosing y = 1/n, it is shown that f_n(1-y) = 0, demonstrating that the function will always obtain its maximum value at x = 1/2.
  • #1
Artusartos
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Homework Statement



Consider [tex]f_n(x) = nx^n(1-x)[/tex] for x in [0,1].

a) What is the limit of [tex]f_n(x)[/tex]?

b) Does [tex]f_n \rightarrow f[/tex] uniformly on [0,1]?

Homework Equations


The Attempt at a Solution



a) 0

b) Yes...

We know that [tex]sup|f_n(x) - f(x)| = |n{\frac{1}{2}}^n(1-\frac{1}{2})|[/tex]...

and

[tex]lim_{n \rightarrow \infty} [sup\{ |f_n(x) - f(x)|: x \in [0,1]\}] = 0[/tex]

So it must be uniformly convergent on [0,1].

Do you think my answer is correct?Thanks in advance
 
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  • #2
Artusartos said:

Homework Statement



Consider [tex]f_n(x) = nx^n(1-x)[/tex] for x in [0,1].

a) What is the limit of [tex]f_n(x)[/tex]?

b) Does [tex]f_n \rightarrow f[/tex] uniformly on [0,1]?



Homework Equations





The Attempt at a Solution



a) 0

b) Yes...

We know that [tex]sup|f_n(x) - f(x)| = |n{\frac{1}{2}}^n(1-\frac{1}{2})|[/tex]...

Why? How do you know they will always obtain their maximum in 1/2??
 
  • #3
If we let [itex]y = 1-x[/itex], then we may write
[tex]f_n(1-y) = n y (1 - y)^n[/tex]
Now what happens if you choose [itex]y = 1/n[/itex]?
 

Related to The title could be: Uniform Convergence of f_n(x) = nx^n(1-x) on [0,1]?

What is uniform convergence?

Uniform convergence is a type of convergence in mathematics, specifically in the context of sequences or series. It means that the sequence or series converges to the same limit at every point in its domain.

How do you test for uniform convergence?

To test for uniform convergence, one can use the Weierstrass M-test, which involves finding a bounding function that can be compared to the original sequence or series. Another method is the Cauchy criterion, which states that for a sequence to be uniformly convergent, the difference between consecutive terms must approach zero as the index approaches infinity.

What is the difference between uniform convergence and pointwise convergence?

The main difference between uniform convergence and pointwise convergence is that in uniform convergence, the sequence or series converges to the same limit at every point in its domain, while in pointwise convergence, the limit may vary at different points.

Why is uniform convergence important?

Uniform convergence is important because it guarantees that the sequence or series converges to the same limit at every point in its domain, which can simplify calculations and make the analysis of functions and equations easier. It is also a fundamental concept in many areas of mathematics, such as analysis, calculus, and differential equations.

What are some common examples of uniformly convergent sequences or series?

Some common examples of uniformly convergent sequences or series include geometric series, power series, and the Taylor series expansion of many functions. Other examples can be found in the convergence of infinite sums, such as the Riemann zeta function, and in the approximation of functions using polynomial or trigonometric series.

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