What is taylor e^x-1 ?

The answer to taylor e^x-1 is x+1/2 x^2+1/6 x^3+1/24 x^4+1/120 x^5+\ldots

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taylor e^x-1

taylor

e^{x} - 1

x + \frac{1}{2} x^{2} + \frac{1}{6} x^{3} + \frac{1}{24} x^{4} + \frac{1}{120} x^{5} + \dots

Solution steps

Apply the Taylor Formula

= 0 + \frac{\frac{d}{d x} ( e ^{x} - 1 ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( e ^{x} - 1 ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( e ^{x} - 1 ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= 0 + \frac{1}{1 !} x + \frac{1}{2 !} x^{2} + \frac{1}{3 !} x^{3} + \frac{1}{4 !} x^{4} + \frac{1}{5 !} x^{5} + \dots

Refine

= x + \frac{1}{2} x^{2} + \frac{1}{6} x^{3} + \frac{1}{24} x^{4} + \frac{1}{120} x^{5} + \dots

x \to 1 lim ((x - 1)^{2})

\frac{\partial}{\partial x} (x^{3} + 2 x y - y^{2})

y = \frac{3}{4 x ^{5}}

x \to 0 lim (\frac{sin ( x )}{6 x})

y = 5 + 4 x^{2} - 2 x^{3}, (2, 5)

What is taylor e^x-1 ?
The answer to taylor e^x-1 is x+1/2 x^2+1/6 x^3+1/24 x^4+1/120 x^5+\ldots