／var／log marcus chiu

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Approximation Theory

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Function Approximation Problem

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Approximating Known Target Functions

Geometric Series／Succession

Created on Sep 13, 2021 · Last Modified on Feb 01, 2024

Geometric Series/Succession

is a type of power series where its coefficients are equal (𝑎) and is centered on zero (𝑐 = 0)
is a type of Maclaurin series where its coefficients are equal (𝑎)
the name indicate each term is the geometric mean of its two neighboring terms

Geometric Series (Finite N)

for 𝑟 ≠ 1:

𝑠 = 𝑎𝑟⁰ + 𝑎𝑟¹ + … + 𝑎𝑟^𝑛-1
𝑠 = 𝛴_{0≤𝑘≤𝑛-1}[𝑎𝑟^𝑘]
𝑠 = 𝑎 [(1 - 𝑟^𝑛) / (1 - 𝑟)]

where:

𝑎 is the first term of the series
𝑟 is the common ratio

formula derivation:

𝑠 = 𝑎 + 𝑎𝑟 + 𝑎𝑟² + … + 𝑎𝑟^𝑛-1
𝑠𝑟 = 𝑎𝑟 + 𝑎𝑟² + … + 𝑎𝑟^𝑛
𝑠 - 𝑠𝑟 = 𝑎 - 𝑎𝑟^𝑛
𝑠(1 - 𝑟) = 𝑎(1 - 𝑟^𝑛)
𝑠 = 𝑎(1 - 𝑟^𝑛) / (1 - 𝑟)

Geometric Series (Finite N)

$\sum_{k = 1}^{n} r^{k} = \frac{r - r ^{n + 1}}{1 - r}$
derivations
- $\sum_{k = 1}^{n} r^{k} = \frac{1 - r ^{n + 1}}{1 - r} - 1$
- $\sum_{k = 1}^{n} r^{k} = \frac{1 - r ^{n + 1}}{1 - r} - \frac{1 - r}{1 - r}$
- $\sum_{k = 1}^{n} r^{k} = \frac{1 - r ^{n + 1} - ( 1 - r )}{1 - r}$
- $\sum_{k = 1}^{n} r^{k} = \frac{1 - r ^{n + 1} - 1 + r}{1 - r}$
- $\sum_{k = 1}^{n} r^{k} = \frac{r - r ^{n + 1}}{1 - r}$

Geometric Series (Infinite)

for 𝑟 < 1:

𝑠 = 𝑎𝑟⁰ + 𝑎𝑟¹ + … + 𝑎𝑟^∞
𝑠 = 𝛴_{0≤𝑘≤∞}[𝑎𝑟^𝑘]
𝑠 = 𝑎/(1-𝑟)

where:

𝑎 is the first term of the series
𝑟 is the common ratio

formula derivation:

𝑠 = 𝑎 + 𝑎𝑟 + 𝑎𝑟² + … + 𝑎𝑟^∞
𝑠𝑟 = 𝑎𝑟 + 𝑎𝑟² + … + 𝑎𝑟^∞
𝑠 - 𝑠𝑟 = 𝑎 - 𝑎𝑟^∞
𝑠(1 - 𝑟) = 𝑎(1 - 𝑟^∞)
𝑠 = 𝑎(1 - 𝑟^∞) / (1 - 𝑟)
𝑠 = 𝑎(1 - 0) / (1 - 𝑟)
𝑠 = 𝑎 / (1 - 𝑟)

Derivatives of Geometric Series (Infinite)

𝛴_{0≤𝑘≤∞}[𝑎𝑟^𝑘]	𝑠	𝑎/(1-𝑟)
𝛴_{1≤𝑘≤∞}[𝑎𝑘𝑟^𝑘-1]	𝑠’	𝑎/(1-𝑟)²
𝛴_{𝑙≤𝑘≤∞}[𝑎𝑘(𝑘-1)(𝑘-2)…(𝑘-𝑙+1)𝑟^𝑘-𝑙]	𝑠^(𝑘-𝑙)	𝑙!𝑎/(1-𝑟)^𝑙+1