26th Derivative of a Function- Power Series

[Justabeginner]

Homework Statement

I cannot write out the equation clearly so I am attaching a file.

Homework Equations

The Attempt at a Solution

sin x= x- x^3/3! + x^5/5! - x^7/7! + ...
sinx/(x)= 1- x^2/3! + x^4/5! - x^6/7! - x^26/27! + ...
(-1)^k x^(2k) / (2k+1)! = g^(2k)(0) x^(2k)/(2k)!
g^(2k)(0)= (-1)^(k)/(2k+1)
g^(2*13)(0)= (-1)^(13)/(26+1)
g^(26)(0)= -1/27

Is this correct? Thank you!

 

[lurflurf]

Yes that is the famous function sinc. Three methods jump out.
$$\text{write} \\
x \, \mathrm{sinc}(x)=\sin(x) \\
\text{then show} \\
\dfrac{d^{27}}{dx^{27}}x \, \mathrm{f}(x) =x \, \mathrm{f}^{(27)}(x)+27 \mathrm{f}^{(26)}(x) \\
\text{write (if you know of integrals)} \\
\mathrm{sinc}(x)=\int_0^1 \! \cos(x \, t) \, \mathrm{d}t \\
\text{then show} \\
\mathrm{sinc}^{(26)}(x)=\int_0^1 \! t^{26} \cos^{(26)}(x \, t) \, \mathrm{d}t \\
\text{write} \\
x \, \mathrm{sinc}(x)=\sin(x) \\
\text{using Maclaurin series as}\\
\sum_{k=0}^\infty \frac{x^{k+1}}{k!}\mathrm{sinc}^{(k)}(x)=\sum_{k=0}^\infty \frac{x^{k}}{k!}\sin^{(k)}(x)\\
\text{then write derivatives of sinc interms of those for sin} \\
\text{It is helpful to remember}\\
\sin^{(n)}(x)=\sin(x+n \, \pi/2)\\
\cos^{(n)}(x)=\cos(x+n \, \pi/2)\\
$$

 

[Justabeginner]

Wow thank you so much. I did not know of that method, nor did I know about the sinc function. I appreciate the new knowledge.