Math Problem Statement

for the numerator: ln ⁡ ( 1 + 4 𝑥 ) ≈ 4 𝑥 − ( 4 𝑥 ) 2 2 + 𝑂 ( 𝑥 3 )

4 𝑥 − 8 𝑥 2 + 𝑂 ( 𝑥 3 ) ln(1+4x)≈4x− 2 (4x) 2

2 𝑥 + 2 𝑥 2 + 𝑂 ( 𝑥 3 ) e 2x −1≈2x+ 2 (2x) 2

6 𝑥 + 12 𝑥 2 + 8 𝑥 3 (1+2x) 3 −1=1+6x+12x 2 +8x 3 −1=6x+12x 2 +8x 3

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Taylor Series
Approximations

Taylor series expansion

Advanced High School

Taylor Series Approximation for ln(x) and e^x - 1

Numerical Approximation of e^(x^2) Using Taylor Series at x=0

Estimating Maximum Error Using Taylor Series Approximation of e^x

Estimating Maximum Error in Approximation of e^x

Taylor Series Expansion of e^-x^2 up to Four Terms