Wikipedia:Reference desk/Archives/Mathematics/2015 January 4

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January 4

Coupon-collector problem with packages of distinct coupons

Is there a name for, or any research on, the variant of the coupon collector's problem where coupons are acquired in batches of constant size (such as packs of trading cards) and duplication within a batch is impossible? NeonMerlin 04:43, 4 January 2015 (UTC)
Corrected your link Rojomoke (talk) 05:09, 4 January 2015 (UTC)

This paper seems to deal with the problem you described. There are many variants of the CCP; I don't know if this particular variant has been named. --Mark viking (talk) 10:46, 4 January 2015 (UTC)

Is every power series representation of a function a Taylor series?

I'm not convinced. We commonly say things along the lines of "$\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}x^{2n}$ is the Taylor series for $e^{-x^{2}}$" but to me, it is not completely trivial to show that this is in accord with the definition of a Taylor series using derivatives and Taylor's theorem. Likewise the Bessel function of the first kind is definable using a power series, but that series results from using the Frobenius method to solve Bessel's equation and I can't see how to derive the same series using Taylor's theorem (otherwise, it would imply that odd derivatives of the function must be zero at the origin while even ones aren't, which seems rather strange to me). The hypergeometric functions aren't readily identifiable as Taylor series to me either. What gives?--Jasper Deng (talk) 09:50, 4 January 2015 (UTC)

Let $f(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}x^{2n}$ and $g(x)=e^{-x^{2}}$. Let $a_{m}={\frac {(-1)^{m/2}}{(m/2)!}}$ if m is even, otherwise 0. Then $f(x)=\sum _{m=0}^{\infty }a_{m}x^{m}$. Since $f(x)=g(x)$ for every x, it follows that $f^{(k)}(x)=g^{(k)}(x)$ for any positive integer k. Assuming we can interchange summation and differentiation (I forgot the exact conditions that guarantee this), we clearly have $f^{(k)}(0)=a_{k}k!$, so $g^{(k)}(0)=a_{k}k!$. So the Taylor series of $g(x)$ is $\sum _{m=0}^{\infty }a_{m}x^{m}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}x^{2n}$.

I'm sure there are theorems that show that, more generally, arithmetic operations and compositions on well-behaved Taylor series "works". -- Meni Rosenfeld (talk) 11:57, 4 January 2015 (UTC)

Yes. In the interior of the radius of convergence, term-by-term differentiation is allowed (by the ratio test). So every power series representation of a function in an open disc centered at x=a is also the Taylor series of the function centered at a. Sławomir Biały (talk) 12:17, 4 January 2015 (UTC)