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superconvergence
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(Definition)
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A sequence $x_0,x_1,\dots$ superconverges to 0 if, when the $x_i$ are written in base 2, then each number $x_i$ starts with $2^i-1\approx 2^i$ zeroes. For example, the following sequence is superconverging to 0.
In this case it is easy to see that the number of binary 0's doubles each $x_n$ .
A sequence $\{x_i\}$ superconverges to $x$ if $\{x_i-x\}$ superconverges to 0, and a sequence $\{y_i\}$ is said to be superconvergent if there exists a $y$ to which the sequence superconverges.
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"superconvergence" is owned by mathcam. [ full author list (2) | owner history (1) ]
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Cross-references: binary, easy to see, number, base, sequence
There is 1 reference to this entry.
This is version 11 of superconvergence, born on 2001-11-13, modified 2004-08-24.
Object id is 793, canonical name is Superconvergence.
Accessed 4351 times total.
Classification:
| AMS MSC: | 41A25 (Approximations and expansions :: Rate of convergence, degree of approximation) |
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Pending Errata and Addenda
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![\begin{displaymath}\begin{array}{clcl} x_{n+1}&=x_n^2 & (x_n)_{10} & (x_n)_2\\ [... ...1.5pt] x_4 &= & \frac{1}{65536} & .0000000000000001 \end{array}\end{displaymath}](http://images.planetmath.org:8080/cache/objects/793/js/img1.png "\begin{displaymath}\begin{array}{clcl} x_{n+1}&=x_n^2 & (x_n)_{10} & (x_n)_2\\ [... ...1.5pt] x_4 &= & \frac{1}{65536} & .0000000000000001 \end{array}\end{displaymath}")