How to Determine Validity of Power Series Representations for s(x) and s'(x)?

[mariama1]

Homework Statement

Find the power series representation for s(x) and s`(x)
integral sin (pi t^2)\2
and which of them is valid ?

Homework Equations

The Attempt at a Solution

I tried to solve this question , but i am not sure
s`(x) = sin (pi t^2)\2 by Abel's theorm

$$\sum\ell^in\theta$$

and it converges
I am not sure about the solution , Is it right ? and what about s(x)
Is it - cos(pi t^2)\2 ? and how we can find the power series representaion for it?

 

[SammyS]

Homework Statement

Find the power series representation for s(x) and s`(x)
I take it that you meant:
$$s(x)=\int_0^x \sin \left(\frac{\pi t^2}{2}\right)\,dt$$

and which of them is valid ?

Homework Equations

The Attempt at a Solution

I tried to solve this question , but i am not sure
s`(x) = [tex]sin (pi t^2)\2[tex] by Abel's theorm
$$\sum$$$$\ell^ in \theta$$
and it converges
I am not sure about the solution , Is it right ? and what about s(x)
Is it $$\ - cos(pi t^2)\2$$ ?

$$s'(x)=\sin \left(\frac{\pi x^2}{2}\right)$$

Do you know the Taylor Series (actually Maclaurin Series) for sin(x) ?

Substitute πx²/2 for x into the Taylor Series for sin(x).

To find the power series for s(x), integrate the power series for s'(t) from 0 (or whatever the original problem had) to x .

 

[mariama1]

$$s'(x)=\sin \left(\frac{\pi x^2}{2}\right)$$

Do you know the Taylor Series (actually Maclaurin Series) for sin(x) ?

Substitute πx²/2 for x into the Taylor Series for sin(x).

To find the power series for s(x), integrate the power series for s'(t) from 0 (or whatever the original problem had) to x .

The taylor serries for sin (x) = $$\sum\frac{-1^{n}}{(2n+1)!}x^{2n+1}$$

and by substituting (pi x^2)\2
we get

$$\sum\frac{-1^{n}}{(2n+1)!}\left(\frac{\pi\times x^{2}}{2}\right)^{2n+1}$$

Woow It is a power series now

s(x) =$$\int \sum\frac{-1^{n}}{(2n+1)!}\left(\frac{\pi\times x^{2}}{2}\right)^{2n+1}$$

but how can i check if these two are valid ?


Thanks a lot for these useful advices

 

[SammyS]

...

s(x) =$$\int \sum\frac{-1^{n}}{(2n+1)!}\left(\frac{\pi\times x^{2}}{2}\right)^{2n+1}$$

but how can i check if these two are valid ?

Thanks a lot for these useful advices

You can actually do the integration if each term in the sum.

$$s(x)=\int \left(\sum_{n=0}^\infty\frac{(-1)^{n}}{(2n+1)!}\left(\frac{\pi\times x^{2}}{2}\right)^{2n+1}\right)\,dx$$

$$=\sum_{n=0}^\infty \left(\int \frac{(-1)^{n}}{(2n+1)!}\left(\frac{\pi\times x^{2}}{2}\right)^{2n+1}\,dx\right)$$

$$=\sum_{n=0}^\infty \left(\frac{(-1)^{n}}{(2n+1)!}\left(\frac{\pi}{2}\right)^{2n+1}\int x^{4n+2}\,dx\right)$$​

To check s'(x), I used WolfrmAlpha.