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Fixed the typo discovered by the commenter.
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Kelvin Lee
Kelvin Lee
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Given the following sum:

$S(n) = \sum_{i=1}^{n} \frac{1}{1-\operatorname{H}(p)}$$S(n) = \sum_{i=1}^{n} \frac{1}{(1-\operatorname{H}(p))^i}$

where $H$ is the binary entropy function defined as:

$\operatorname{H}(p) = -p\log p - (1-p)\log (1-p) $.

Let $f(n) = \frac{n}{S(n)}$.

Assume $p$ is very small, is it possible to approximate the $S(n)$ and $f(n)$ defined above with simpler closed formulae e.g. a polynomial of $p$ without loosing much precision?

Given the following sum:

$S(n) = \sum_{i=1}^{n} \frac{1}{1-\operatorname{H}(p)}$

where $H$ is the binary entropy function defined as:

$\operatorname{H}(p) = -p\log p - (1-p)\log (1-p) $.

Let $f(n) = \frac{n}{S(n)}$.

Assume $p$ is very small, is it possible to approximate the $S(n)$ and $f(n)$ defined above with simpler closed formulae e.g. a polynomial of $p$ without loosing much precision?

Given the following sum:

$S(n) = \sum_{i=1}^{n} \frac{1}{(1-\operatorname{H}(p))^i}$

where $H$ is the binary entropy function defined as:

$\operatorname{H}(p) = -p\log p - (1-p)\log (1-p) $.

Let $f(n) = \frac{n}{S(n)}$.

Assume $p$ is very small, is it possible to approximate the $S(n)$ and $f(n)$ defined above with simpler closed formulae e.g. a polynomial of $p$ without loosing much precision?

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Kelvin Lee
Kelvin Lee
  • 285
  • 2
  • 8

Approximation of the sum involving binary entropy function

Given the following sum:

$S(n) = \sum_{i=1}^{n} \frac{1}{1-\operatorname{H}(p)}$

where $H$ is the binary entropy function defined as:

$\operatorname{H}(p) = -p\log p - (1-p)\log (1-p) $.

Let $f(n) = \frac{n}{S(n)}$.

Assume $p$ is very small, is it possible to approximate the $S(n)$ and $f(n)$ defined above with simpler closed formulae e.g. a polynomial of $p$ without loosing much precision?