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infinite_geometric_series [2016/07/10 22:01] nikolaj old revision restored (2016/07/10 17:44) |
infinite_geometric_series [2019/09/23 21:19] nikolaj |
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| $\sum_{k=0}^\infty\left(1-\dfrac{1}{z}\right)^kX^k = z+(X-1)\left(1+\dfrac{1}{z}\right)\dfrac{1}{z+(1-X)}$ | $\sum_{k=0}^\infty\left(1-\dfrac{1}{z}\right)^kX^k = z+(X-1)\left(1+\dfrac{1}{z}\right)\dfrac{1}{z+(1-X)}$ | ||
| - | == Notes == | + | == q-Integral == |
| + | For a function $f$, the q-integral from $0$ to $1$ ("$z$-integral" if we stick to our notation above) is defined as | ||
| + | |||
| + | $\sum_{k=0}^\infty f(z^k)\,z^k=\dfrac{1}{1-z}\cdot\int_0^1 f(s)\,{\mathrm d}_zs$ | ||
| + | |||
| + | == Related notes == | ||
| $z = \sum_{k=0}^\infty\left(z^{-1}(z-1)\right)^k = \sum_{k=0}^\infty\left(1-z^{-1}\right)^k = \sum_{k=0}^\infty \sum_{m=0}^k {k \choose m}(-z)^{-m} $ | $z = \sum_{k=0}^\infty\left(z^{-1}(z-1)\right)^k = \sum_{k=0}^\infty\left(1-z^{-1}\right)^k = \sum_{k=0}^\infty \sum_{m=0}^k {k \choose m}(-z)^{-m} $ | ||
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| $ \sum_{k=0}^{n-1} \sum_{m=0}^k {k \choose m}(-z)^{-m} = z\left(1-\left(\dfrac{z-1}{z}\right)^n\right)$ | $ \sum_{k=0}^{n-1} \sum_{m=0}^k {k \choose m}(-z)^{-m} = z\left(1-\left(\dfrac{z-1}{z}\right)^n\right)$ | ||
| - | We can get rid of the double sum as follows: | + | See also [[Niemand sequences]]. |
| - | + | ||
| - | $ = \sum_{k=0}^{n-1}\left(\prod_{j=0}^k\dfrac{n-j}{1+j}\right)(-z)^{-k} $ | + | |
| - | + | ||
| - | But note that here the summands depends on the upper bound of the sum and for this we can't take the limit $n\to \infty$ of the summand term anymore. | + | |
| - | + | ||
| - | There are some related sums, e.g. | + | |
| + | == Powers == | ||
| <code> | <code> | ||
| - | expFactor[n_] = (n - (k - j))/(j*f[n]); | + | Sum[Binomial[-s, k] x^k, {k, 0, \[Infinity]}] |
| - | invFactor[n_] = -(n - j)/(j + 1); | + | Series[(1 - x)^-s, {x, 0, 4}] |
| - | Sum[Product[expFactor[n]*z, {j, 1, k}], {k, 0, n}] // Simplify | + | Binomial[-s, 3]; |
| - | Sum[Product[invFactor[n]*z, {j, 1, k}], {k, 0, n}] // Simplify | + | % - (-s)!/(3! ((-s) - 3)!) // FullSimplify |
| - | </code> | + | %% + 1/6 (2 s + 3 s^2 + s^3) // FullSimplify |
| - | + | ||
| - | or written differently | + | |
| - | + | ||
| - | <code> | + | |
| - | Sum[Product[1 - (k - j)/n, {j, 1, k}] z^k/k!, {k, 0, n}] // Simplify | + | |
| - | + | ||
| - | Sum[Product[-(n - j), {j, 1, k}] (1/(k + 1)) z^k/k!, {k, 0, n}] // Simplify | + | |
| </code> | </code> | ||
