Author: Rishabh Kumar (IIT + ISI | Global Math Mentor)
Published: October 2025
Category: Calculus | Series Expansion | Advanced Concepts
🔹 What Is a Taylor Series?
The Taylor Series is one of the most elegant ideas in mathematics — it allows us to approximate any smooth function using polynomials.
If you’ve ever used your calculator to compute sin(x)\sin(x)sin(x), exe^xex, or ln(1+x)\ln(1+x)ln(1+x), you’ve relied on Taylor expansions behind the scenes.
Taylor Series bridges algebra and calculus — turning complex functions into infinitely long polynomials that behave almost exactly like the original.
🔹 The General Formula
If f(x)f(x)f(x) is infinitely differentiable near x=ax = ax=a, then its Taylor Series expansion about x=ax = ax=a is: f(x)=f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+f′′′(a)3!(x−a)3+⋯f(x) = f(a) + f'(a)(x – a) + \frac{f”(a)}{2!}(x – a)^2 + \frac{f”'(a)}{3!}(x – a)^3 + \cdotsf(x)=f(a)+f′(a)(x−a)+2!f′′(a)(x−a)2+3!f′′′(a)(x−a)3+⋯
or compactly: f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x – a)^nf(x)=n=0∑∞n!f(n)(a)(x−a)n
When a=0a = 0a=0, this becomes the Maclaurin Series. f(x)=f(0)+f′(0)x+f′′(0)2!x2+f′′′(0)3!x3+⋯f(x) = f(0) + f'(0)x + \frac{f”(0)}{2!}x^2 + \frac{f”'(0)}{3!}x^3 + \cdotsf(x)=f(0)+f′(0)x+2!f′′(0)x2+3!f′′′(0)x3+⋯
🔹 Step-by-Step Example — exe^xex Expansion
Let’s find the Maclaurin Series for f(x)=exf(x) = e^xf(x)=ex. f(x)=ex,f′(x)=ex,f′′(x)=ex,…f(x) = e^x, \quad f'(x) = e^x, \quad f”(x) = e^x, \ldotsf(x)=ex,f′(x)=ex,f′′(x)=ex,…
At x=0x = 0x=0: f(n)(0)=1f^{(n)}(0) = 1f(n)(0)=1 for all nnn. ex=1+x+x22!+x33!+x44!+⋯e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdotsex=1+x+2!x2+3!x3+4!x4+⋯
✅ exe^xex is its own series!
That’s why exponential functions appear everywhere — they’re infinitely differentiable and perfectly approximated by their Taylor expansion.
🔹 Example 2 — sin(x)\sin(x)sin(x) Expansion
f(x)=sinx,f′(x)=cosx,f′′(x)=−sinx,f′′′(x)=−cosxf(x) = \sin x, \quad f'(x) = \cos x, \quad f”(x) = -\sin x, \quad f”'(x) = -\cos xf(x)=sinx,f′(x)=cosx,f′′(x)=−sinx,f′′′(x)=−cosx
At x=0x = 0x=0: f(0)=0, f′(0)=1, f′′(0)=0, f′′′(0)=−1f(0) = 0, \ f'(0) = 1, \ f”(0) = 0, \ f”'(0) = -1f(0)=0, f′(0)=1, f′′(0)=0, f′′′(0)=−1 sinx=x−x33!+x55!−x77!+⋯\sin x = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdotssinx=x−3!x3+5!x5−7!x7+⋯
✅ This infinite polynomial perfectly models the sine curve — smooth, oscillating, and symmetric.
🔹 Example 3 — ln(1+x)\ln(1+x)ln(1+x) Expansion
f(x)=ln(1+x),f′(x)=11+x, f′′(x)=−1(1+x)2f(x) = \ln(1+x), \quad f'(x) = \frac{1}{1+x}, \ f”(x) = -\frac{1}{(1+x)^2}f(x)=ln(1+x),f′(x)=1+x1, f′′(x)=−(1+x)21
At x=0x = 0x=0: ln(1+x)=x−x22+x33−x44+⋯\ln(1+x) = x – \frac{x^2}{2} + \frac{x^3}{3} – \frac{x^4}{4} + \cdotsln(1+x)=x−2×2+3×3−4×4+⋯
Valid for ∣x∣<1|x| < 1∣x∣<1.
🔹 Why Does It Work?
The Taylor Series builds the function by matching its value and derivatives at one point.
The more terms you take, the more accurate the approximation near x=ax = ax=a.
In physics and engineering, even a 2-term or 3-term expansion often provides enough precision for practical modeling.
🔹 Applications of Taylor Series
- Approximating functions when calculators are not allowed (STEP/MAT)
- Physics: small-angle approximations (sinx≈x\sin x \approx xsinx≈x)
- Economics: growth modeling with exponential approximations
- Computer science: algorithm design, numerical analysis
- Mathematical reasoning: local linearization and error estimation
🔹 Common Pitfalls
- ❌ Forgetting factorial denominators (n!n!n!).
- ❌ Mixing up signs in alternating series like sinx\sin xsinx, ln(1+x)\ln(1+x)ln(1+x).
- ❌ Expanding around wrong point (a≠0a \neq 0a=0) — always check the center.
- ❌ Ignoring radius of convergence — not all series work for all xxx.
🔹 Advanced Insight — Radius of Convergence
A Taylor Series may not converge for all xxx.
Use the ratio test: limn→∞∣an+1an∣=L\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = Ln→∞limanan+1=L
If L<1L < 1L<1, the series converges; if L>1L > 1L>1, it diverges.
For example, ln(1+x):∣x∣<1(convergent)\ln(1+x): |x| < 1 \quad \text{(convergent)}ln(1+x):∣x∣<1(convergent) ex:convergent for all xe^x: \text{convergent for all } xex:convergent for all x
🌟 Why Taylor Series Matters
Taylor Series shows how local information (derivatives at one point) can rebuild entire functions — one of the deepest ideas in calculus.
It forms the backbone of:
- Higher-level pure math,
- Differential equations,
- Mathematical modeling,
- and university entrance tests (STEP, MAT, TMUA).
📘 Learn Beyond Memorization
At Math By Rishabh, Taylor Series is taught as a story — from intuition to rigorous understanding.
In the Mathematics Elevate Mentorship Program, you’ll:
✅ Learn to derive expansions from first principles,
✅ Solve STEP/MAT-style series problems,
✅ Build analytical fluency across calculus and beyond.
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