Approximate square function

  • #1

Homework Statement:

Is there some way to approximate this function, for x=1?

Relevant Equations:

The function is ##(x-1)^{1/n}##, with ##n## integer and n>1 and n=odd.
Obviously, a priori it is not possible tu use the Taylor series because the derivative ##\sim (x-1)^{1/n-1}## is not well defined in x=1.

Is there any mathematical trick? or, other approximation?
 
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  • #2
PeroK
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Homework Statement:: Is there some way to approximate this function, for x=1?
Relevant Equations:: The function is ##(x-1)^{1/n}##, with ##n## integer and n>1 and n=odd.

Obviously, a priori it is not possible tu use the Taylor series because the derivative ##\sim (x-1)^{1/n-1}## is not well defined in x=1.

Is there any mathematical trick? or, other approximation?
Do you mean for ##x \approx 1##?

At ##x = 1## you have ##0^{1/n} = 0##.
 
  • #3
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[tex]f'(x_0) = \lim _{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0} \Rightarrow f'(x_0)(x-x_0) +f(x_0) = f(x) +o(|x-x_0|),\ x\to x_0 .[/tex]
Take [itex]x_0 =1 + \delta[/itex]. If [itex]|x-x_0|[/itex] is small, then it's a pretty good linear approximation.
 
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  • #4
Thanks. I dont understand your idea, beacause ##f'(x_0)## does not exist for x_0=1
 
  • #5
Do you mean for ##x \approx 1##?

At ##x = 1## you have ##0^{1/n} = 0##.
Thanks. This is my doubt.

As for example, if is there any approximation of ##x^{1/3}## near x=0, where Taylor is not possible because the function is not differentiable
 
  • #6
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Thanks. I dont understand your idea, beacause ##f'(x_0)## does not exist for x_0=1
Indeed, but if you can calculate value of derivative exactly just a bit further from [itex]1[/itex], then you could approximate linearly around a very small neighborhood of [itex]x_0>1[/itex].
 
  • #7
PeroK
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Thanks. This is my doubt.

As for example, if is there any approximation of ##x^{1/3}## near x=0, where Taylor is not possible because the function is not differentiable
If ##x## is small, then powers of ##x## are smaller. But, ##x^{1/3} > x##, so you are not going to get a power series approximation to ##x## in powers of ##x##.
 
  • #8
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For the record, non-smooth approximation is hard. Most of the theory I've come across always assumes differentiability or absolute continuity or something "nice". Non-smooth continuous functions need not be nice at all.
 
  • #9
Indeed, but if you can calculate value of derivative exactly just a bit further from [itex]1[/itex], then you could approximate linearly around a very small neighborhood of [itex]x_0>1[/itex].
I dont understand :(
If ##x## is small, then powers of ##x## are smaller. But, ##x^{1/3} > x##, so you are not going to get a power series approximation to ##x## in powers of ##x##.
This mean that, in my problem ##(1-x)^{1/n}## is not possible to approximate the function for ##x_0 \approx 1##?
 
  • #10
PeroK
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This mean that, in my problem ##(1-x)^{1/n}## is not possible to approximate the function for ##x_0 \approx 1##?
I don't see how you can do it with a Taylor series. One problem is that the derivatives are unbounded near ##x = 1##.
 
  • #11
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For an approximation, I suggest you expand using the binomial theorem and take the limit as x approaches 1. For ##x\gt 1## $$(x-1)^{\frac{1}{n}}=(x)^{\frac{1}{n}}(1-\frac{1}{x})^{\frac{1}{n}}=(x)^{\frac{1}{n}}\sum_{k=0}^{\infty} \begin{pmatrix}
\frac{1}{n} \\
k
\end{pmatrix} (\frac{1}{x})^k $$Use the identity$$
\begin{pmatrix}
\frac{1}{n} \\
k
\end{pmatrix}=\frac{(-1)^k}{k!} \prod_{j=0}^{k-1} (j-\frac{1}{n})$$ for each term in the series.
 
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  • #12
haruspex
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You might as well work in terms of ##x^{\frac 1n}##, for x near zero,
It depends what you want from an approximation. Generally you want such as an easy way to generate numerical solutions. What form will x be in? If in scientific notation, ##x=a\times 10^m##, you have ##a^{\frac 1n}\times 10^{\frac mn}##. For large n you can ignore ##a^{\frac 1n}##, and it’s easy to reduce m so that it is less than n.
This leaves you with ##10^y##, 0<y<1. For that, you can use ##e^{y\ln(10)}## or ##e^{y\ln(10)+\ln(a)}##.

But is the problem as stated in post #1 as it was given to you? If not, please post the entire original problem.
 

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