∑Taylor Series Calculator
A Taylor series approximates any smooth function as an infinite polynomial around a point. Truncating it gives a polynomial approximation — the basis for how calculators compute sin, cos, and eˣ.
- 1f(x) ≈ f(a) + f'(a)(x−a) + f''(a)(x−a)²/2! + ...
- 2sin(x) = x − x³/6 + x⁵/120 − ...
- 3cos(x) = 1 − x²/2 + x⁴/24 − ...
- 4eˣ = 1 + x + x²/2 + x³/6 + ...
sin(0.5 rad) using 3 terms=0.5 − 0.5³/6 + 0.5⁵/120 = 0.4794Exact: 0.47943; error < 0.0001
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Fun Fact
Taylor series explain why CPUs can compute transcendental functions using only addition and multiplication — they evaluate polynomial approximations in hardware.
References
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