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What is Taylor Series?
The Taylor series, also called the Taylor expansion, expresses a smooth function f(x) as an infinite polynomial around a center a.
Taylor Series formula
The general Taylor expansion of f(x) about point a is:
f(x) = Σₙ₌₀⁽∞⁾ [ f⁽ⁿ⁾(a) / n! ] · (x – a)ⁿ
How to solve Taylor Series step by step
Step 1: Identify your function f(x) and choose the expansion point a (commonly 0).
Step 2: Compute the derivatives f(a), f′(a), f″(a), … up to the order you need.
Step 3: Plug each derivative into the term (f⁽ⁿ⁾(a) / n!) · (x – a)ⁿ.
Step 4: Truncate the infinite sum at n = m for a practical polynomial approximation.
Example: eˣ about a = 0
Since every derivative of eˣ at 0 equals 1:
eˣ = Σₙ₌₀⁽∞⁾ (xⁿ / n!) = 1 + x + x²/2! + x³/3! + ⋯
Tips & Tricks
- Check the radius of convergence to know for which
xvalues the expansion holds. - Memorize common Taylor expansions like
sin x,cos x, andln(1+x)for quick recall. - Balance the number of terms against computation cost for optimal accuracy.
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