Approximating Known Target Functions

Taylor Series vs Taylor Polynomial

is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point

Maclaurin Series vs Maclaurin Polynomial

is a special case of the Taylor Polynomial, that uses zero as our single point

Geometric Series
Geometric Progression

𝑎 [(1 - 𝑟^𝑛) / (1 - 𝑟)] = 𝛴_{0≤𝑘≤𝑛-1}[𝑎𝑟^𝑘] # for 𝑟 ≠ 1
𝑎/(1-𝑟) = 𝛴_{0≤𝑖≤∞}[𝑎𝑟^𝑖] # for |𝑟| < 1

• 1/(1-𝑟) = 𝛴_{0≤𝑖≤∞}[𝑟^𝑖] # for |𝑟| < 1
• 𝑟/(1-𝑟)= 𝛴_{0≤𝑖≤∞}[𝑟^𝑖+1] # for |𝑟| < 1
• 𝑟/(1-𝑟)²= 𝛴_{0≤𝑖≤∞}[𝑖𝑟^𝑖] # for |𝑟| < 1

Padé Approximants

• is the “best” approximation of a function near a specific point by a rational function of a given order

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Fourier Analysis

• Fourier Series - is a way of representing a periodic function as a (possibly infinite) sum of sine and cosine functions
• Fourier Transform - is a way of representing a non-periodic function (satisfying some constraints) as a (possibly infinite) sum of sine and cosine functions
• etc