Alt. approach to Taylor series of derivative of arcsin(x)?

In summary, there was a discussion about using a trigonometric substitution to find the Taylor Series expansion of f(x) = 1/sqrt(1-x^2). The suggested substitution, x = sin(u), did not yield the correct result due to incorrect handling of differentials. Alternative perspectives and confirmation were requested from the community.
  • #1
raxAdaam
32
0
Hi there,

I was hammering out the coefficients for the Taylor Series expansion of [itex]f(x) = \frac{1}{\sqrt{1-x^2}}[/itex], which proved to be quite unsatisfying, so decide to have a look around online for alt. approaches.

What I found (in addition to the method that uses the binomial theorem) was an old post here claiming that a trig sub could be used (here, post #5). However, the sub. - at least as written there, definitely doesn't line up:

[itex]x = sin(u)[/itex], which yields [itex]f(x) = sec(u)[/itex]. However the post seems to mix up the differentials, claiming: [itex]\frac{df}{dx} = \frac{df}{du}\cdot\frac{dx}{du}[/itex], which is - unless I'm entirely missing something - not correct, right? We should actually have:

[itex]\frac{df}{dx} = \frac{df}{du}\cdot\frac{du}{dx} = sec(u)tan(u)\cdot \frac{1}{\sqrt{1-x^2}} = sec(u)tan(u)\cdot\frac{1}{cos(u)}[/itex] ...

I'm hoping that I've missed something here, because the method would be stellar and super insightful if it was valid, but it seems the differentials are not handled properly - can anyone confirm this?


Cheers,

Rax
 
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  • #2
Many views - any thoughts?!

Would be very appreciative of a confirmation from anyone and/or thoughts/perspectives that might lend alternative insight.

Kind regards,


Rax
 
  • #3
You're right.
 

Related to Alt. approach to Taylor series of derivative of arcsin(x)?

1. What is an alternative approach to finding the Taylor series of the derivative of arcsin(x)?

An alternative approach to finding the Taylor series of the derivative of arcsin(x) is by using the power series representation of the derivative of arcsin(x) and then manipulating it to obtain the desired Taylor series. This approach involves using the chain rule and the power series representation of the derivative of arcsin(x).

2. How is this alternative approach different from the traditional approach?

The traditional approach involves using the definition of the derivative, evaluating it at x=0 and then using the general formula for the Taylor series to obtain the series. In contrast, the alternative approach uses the power series representation of the derivative of arcsin(x) and manipulates it to obtain the Taylor series.

3. What are the advantages of using this alternative approach?

The alternative approach can be easier to use and require less computation compared to the traditional approach. It also provides a more direct and intuitive way of obtaining the Taylor series of the derivative of arcsin(x).

4. Are there any limitations to this alternative approach?

The alternative approach may not be applicable in all situations, especially when the power series representation of the derivative of arcsin(x) is not readily available. It also requires a good understanding of the chain rule and power series manipulation.

5. Can this alternative approach be extended to other trigonometric functions?

Yes, this alternative approach can be extended to other trigonometric functions such as arccos(x), arctan(x), etc. by using the corresponding power series representation of their derivatives and applying the same manipulation techniques.

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