PlanetMath: extracting every $n^\mathrm{th}$ term of a series

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extracting every $n^\mathrm{th}$ term of a series

(Theorem)

Roots of unity can be used to extract every $n^\mathrm{th}$ term of a series. This method is due to Simpson [1759].

Theorem. Let $\omega=e^{2\pi i/k}$ be a primitive $k^\mathrm{th}$ root of unity. If $f(x)=\sum_{j=0}^{\infty} a_j x^j$ and $n\not\equiv 0\pmod k$ , then

$\displaystyle \sum_{j=0}^{\infty} a_{kj+n}x^{kj+n}=\frac{1}{k}\sum_{j=0}^{k-1}\omega^{-jn}f(\omega^{j}x)$

Proof. This is a consequence of the fact that $\sum_{j=0}^{k-1}\omega^{jm}=0$ for $m\not\equiv 0\pmod k$ .

Consider the term involving on the right-hand side. It is

$\displaystyle \frac{1}{k}\sum_{j=0}^{k-1}\omega^{-jn}a_r\omega^{jr}x^r=\frac{1}{k}a_rx^r\sum_{j=0}^{k-1}\omega^{j(r-n)}$

If $r\not\equiv n\pmod k$ , the sum is zero. So the term involving

is zero unless $r\equiv n\pmod k$ , in which case it is

since each element of the sum is

Note that this method is a generalization of the commonly known trick for extracting alternate terms of a series:

$\displaystyle \frac{1}{2}(f(x)-f(-x))$

produces the odd terms of

"extracting every $n^\mathrm{th}$ term of a series" is owned by rm50.

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Cross-references: odd, sum, side, consequence, primitive, series, term, roots of unity
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This is version 7 of extracting every $n^\mathrm{th}$ term of a series, born on 2006-11-11, modified 2006-11-12.
Object id is 8538, canonical name is ExtractingEveryNthTermOfASeries.
Accessed 497 times total.

Classification:

AMS MSC:

11-00 (Number theory :: General reference works )

Pending Errata and Addenda

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