Math Problem Statement
Over the given interval I determine the sets of uniform convergence for: f n (x)= sin(nx) n
√
,I=R.
Question 1 Select one:
a. f ⟶ uniform 0 on [0,1].
b. f ⟶ uniform 0 on R.
c. f ⟶ uniform 0 on [0,∞).
d. f ⟶ uniform 0 on
Solution
Let's analyze the sequence of functions over the interval and determine the set on which it converges uniformly.
Step 1: Pointwise Convergence
First, let's determine the pointwise limit of as .
Given:
As increases, oscillates between and , but the amplitude of the function is scaled by . This means that: for all .
So, converges pointwise to on .
Step 2: Uniform Convergence
Next, we need to check whether this convergence is uniform over different intervals.
To determine uniform convergence, consider the supremum of over any interval :
Since for all and , we have:
As , approaches 0. This indicates that converges uniformly to 0 over any bounded interval in .
However, for unbounded intervals like or , the supremum over could potentially involve values of where does not behave uniformly well, so we need to be more careful.
Step 3: Analyze Options
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Option a: converges uniformly to 0 on .
- On the interval , which is bounded, we can apply our earlier result. uniformly converges to 0 as increases.
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Option b: converges uniformly to 0 on .
- On the entire real line, could be problematic because could be large, causing oscillations that do not die out uniformly. Hence, this option is less likely to be true.
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Option c: converges uniformly to 0 on .
- Similar to the argument above for the entire real line, the interval is unbounded, and uniform convergence over an unbounded interval is generally more challenging to establish.
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Option d: converges uniformly to 0 on some unspecified set.
- This is too vague to be a valid answer.
Conclusion
The correct choice is:
a. converges uniformly to 0 on .
Would you like further details or have any questions?
Here are five relative questions to expand your understanding:
- How do we determine the pointwise limit of a sequence of functions?
- What is the difference between pointwise convergence and uniform convergence?
- Why is uniform convergence more difficult to achieve on unbounded intervals?
- Can you think of an example where a function converges pointwise but not uniformly?
- How does the behavior of impact the convergence properties of ?
Tip: Uniform convergence implies that the functions in the sequence get uniformly close to the limit function, regardless of the specific point in the domain.
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Math Problem Analysis
Mathematical Concepts
Uniform convergence
Pointwise convergence
Supremum
Formulas
$f_n(x) = \frac{\sin(nx)}{\sqrt{n}}$
Theorems
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Suitable Grade Level
Advanced undergraduate