Understanding the Convergence of Radical Functions

In summary, when subtracting radical functions, it is important to simplify each radical individually and then combine like terms. Different types of radicals can be subtracted as long as they have the same index and radicand. When subtracting radicals with different indices, rewrite them with the same index first. Radicals with coefficients can also be subtracted by distributing the coefficient and following the same steps. To ensure that the radical is simplified correctly, you can rewrite it back to its original form or use a calculator to compare the results.
  • #1
mathvision
5
0
This is something that comes up when I want to determine whether the sequence of functions {f_n} converge uniformly to f:

Suppose f_n(x) = sqrt(x^2 + 1/n^2), so f(x) = x.
Then, according to Spivak, f(x) - f_n(x) = sqrt(x^2) - sqrt(x^2 + 1/n^2) = 1/(2n^2*sqrt(ε)) for some ε such that x^2 < ε < x^2 + 1/n^2.

Similarly, sqrt(x) - sqrt(x + 1/n) = 1/(2n sqrt(ε)) for some ε such that x < ε < x + 1/n.

Why is this?

I'd really appreciate any help. Thanks!
 
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  • #2
Never mind--figured it out.
 

Related to Understanding the Convergence of Radical Functions

1. How do you subtract radical functions?

To subtract radical functions, you must first simplify each radical separately. Then, combine like terms by adding or subtracting the coefficients of the simplified radicals. Finally, if there are any like radicals, you can combine them by following the rules of radical operations.

2. Can you subtract different types of radicals?

Yes, you can subtract different types of radicals as long as they have the same index and radicand. For example, you can subtract √x from √2x, but you cannot subtract √x from √x2.

3. What is the rule for subtracting radicals with different indices?

The rule for subtracting radicals with different indices is to first rewrite the radicals with the same index by using the properties of radicals. Then, you can follow the same steps as subtracting radicals with the same index.

4. Can you subtract radicals with coefficients?

Yes, you can subtract radicals with coefficients. Simply distribute the coefficient to each term inside the radical and then follow the steps for subtracting radicals with the same index.

5. How do you know if you simplified the radical correctly before subtracting?

You can check if you simplified the radical correctly by rewriting the simplified radical back into its original form and seeing if it matches the original radical. Additionally, you can use a calculator to simplify and compare the results.

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