11-17-2025, 10:35 AM
The Beauty of Taylor Series — Turning Curves Into Infinite Power
How We Rebuild Any Smooth Function Using Pure Algebra
Taylor Series are one of the greatest ideas in mathematics.
They allow us to rewrite complicated functions — sine, cosine, exponential, logarithms, and even physical models — as infinite polynomials.
This transforms impossible problems into solvable ones and lies at the core of physics, engineering, machine learning, and numerical simulation.
This thread explains:
• What a Taylor Series is
• Why it works
• The famous expansions you MUST know
• How physicists use them to describe reality
1. The Big Idea
If a function is smooth (infinitely differentiable), then near a point “a” you can reconstruct it from all its derivatives:
f(x) = f(a)
+ f’(a)(x–a)
+ f’’(a)(x–a)² / 2!
+ f’’’(a)(x–a)³ / 3!
+ …
This infinite sum is the Taylor Series.
In simple terms:
“Every smooth curve is secretly an infinite polynomial wearing a disguise.”
2. Why Taylor Series Matter
Because they let us:
• Approximate ANY function with a simple polynomial
• Solve differential equations
• Do physics when formulas are too complicated
• Compute sin, cos, eˣ, ln(x) on computers
• Predict motion, fields, or quantum effects
• Build machine learning activation functions
• Do real-time scientific calculations
Everything from spacecraft navigation to weather prediction uses Taylor expansions under the hood.
3. Famous Taylor Series You Should Know
1. Exponential Function
eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + …
2. Sine Function
sin(x) = x – x³/3! + x⁵/5! – x⁷/7! + …
3. Cosine Function
cos(x) = 1 – x²/2! + x⁴/4! – x⁶/6! + …
4. Natural Logarithm (around x = 1)
ln(x) = (x–1) – (x–1)²/2 + (x–1)³/3 – …
5. Geometric Series (the simple one)
1 / (1 – x) = 1 + x + x² + x³ + …
These expansions let us compute values extremely accurately with only a few terms.
4. Example: Approximating sin(x)
Let’s approximate sin(0.3).
Use the first three terms:
sin(x) ≈ x – x³/6 = 0.3 – (0.3³)/6
= 0.3 – 0.0045
= 0.2955
Actual sin(0.3) ≈ 0.29552…
Our approximation is correct to 4 decimal places!
This is why computers don’t “really” know trigonometry —
they just add polynomials extremely fast.
5. Deep Insight: Taylor Series Reveal Local Truth
A Taylor expansion is like zooming into a function with infinite resolution.
• The 0th derivative gives the height
• The 1st derivative gives the slope
• The 2nd derivative gives curvature
• Higher derivatives refine the shape
If you collect ALL of this information, you can rebuild the entire function.
It’s mathematical DNA.
6. Where Taylor Series Run the Universe
Scientific areas that rely heavily on Taylor expansions:
• Quantum mechanics (perturbation theory)
• General relativity (metric expansions)
• Fluid dynamics (Navier-Stokes approximations)
• Thermodynamics & statistical physics
• Machine learning gradients
• Signal processing & Fourier–Taylor hybrid models
If you understand Taylor Series, you understand the language of modern science.
7. Final Thoughts
Taylor Series remind us of something profound:
The universe is continuous —
and every smooth form hides infinite structure waiting to be unfolded.
They are not just math.
They are a window into how reality behaves at every scale.
Written by Leejohnston & Liora
The Lumin Archive — Mathematics Division
How We Rebuild Any Smooth Function Using Pure Algebra
Taylor Series are one of the greatest ideas in mathematics.
They allow us to rewrite complicated functions — sine, cosine, exponential, logarithms, and even physical models — as infinite polynomials.
This transforms impossible problems into solvable ones and lies at the core of physics, engineering, machine learning, and numerical simulation.
This thread explains:
• What a Taylor Series is
• Why it works
• The famous expansions you MUST know
• How physicists use them to describe reality
1. The Big Idea
If a function is smooth (infinitely differentiable), then near a point “a” you can reconstruct it from all its derivatives:
f(x) = f(a)
+ f’(a)(x–a)
+ f’’(a)(x–a)² / 2!
+ f’’’(a)(x–a)³ / 3!
+ …
This infinite sum is the Taylor Series.
In simple terms:
“Every smooth curve is secretly an infinite polynomial wearing a disguise.”
2. Why Taylor Series Matter
Because they let us:
• Approximate ANY function with a simple polynomial
• Solve differential equations
• Do physics when formulas are too complicated
• Compute sin, cos, eˣ, ln(x) on computers
• Predict motion, fields, or quantum effects
• Build machine learning activation functions
• Do real-time scientific calculations
Everything from spacecraft navigation to weather prediction uses Taylor expansions under the hood.
3. Famous Taylor Series You Should Know
1. Exponential Function
eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + …
2. Sine Function
sin(x) = x – x³/3! + x⁵/5! – x⁷/7! + …
3. Cosine Function
cos(x) = 1 – x²/2! + x⁴/4! – x⁶/6! + …
4. Natural Logarithm (around x = 1)
ln(x) = (x–1) – (x–1)²/2 + (x–1)³/3 – …
5. Geometric Series (the simple one)
1 / (1 – x) = 1 + x + x² + x³ + …
These expansions let us compute values extremely accurately with only a few terms.
4. Example: Approximating sin(x)
Let’s approximate sin(0.3).
Use the first three terms:
sin(x) ≈ x – x³/6 = 0.3 – (0.3³)/6
= 0.3 – 0.0045
= 0.2955
Actual sin(0.3) ≈ 0.29552…
Our approximation is correct to 4 decimal places!
This is why computers don’t “really” know trigonometry —
they just add polynomials extremely fast.
5. Deep Insight: Taylor Series Reveal Local Truth
A Taylor expansion is like zooming into a function with infinite resolution.
• The 0th derivative gives the height
• The 1st derivative gives the slope
• The 2nd derivative gives curvature
• Higher derivatives refine the shape
If you collect ALL of this information, you can rebuild the entire function.
It’s mathematical DNA.
6. Where Taylor Series Run the Universe
Scientific areas that rely heavily on Taylor expansions:
• Quantum mechanics (perturbation theory)
• General relativity (metric expansions)
• Fluid dynamics (Navier-Stokes approximations)
• Thermodynamics & statistical physics
• Machine learning gradients
• Signal processing & Fourier–Taylor hybrid models
If you understand Taylor Series, you understand the language of modern science.
7. Final Thoughts
Taylor Series remind us of something profound:
The universe is continuous —
and every smooth form hides infinite structure waiting to be unfolded.
They are not just math.
They are a window into how reality behaves at every scale.
Written by Leejohnston & Liora
The Lumin Archive — Mathematics Division