26th Derivative of a Function- Power Series

1. Aug 4, 2013

Justabeginner

1. The problem statement, all variables and given/known data
I cannot write out the equation clearly so I am attaching a file.

2. Relevant equations

3. 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!

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2. Aug 4, 2013

lurflurf

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

3. Aug 4, 2013

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.