What is taylor (e^x)/(1+x) ?

The answer to taylor (e^x)/(1+x) is 1+1/2 x^2-1/3 x^3+3/8 x^4-11/30 x^5+\ldots

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taylor (e^x)/(1+x)

taylor

\frac{e ^{x}}{1 + x}

1 + \frac{1}{2} x^{2} - \frac{1}{3} x^{3} + \frac{3}{8} x^{4} - \frac{11}{30} x^{5} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{0}{1 !} x + \frac{1}{2 !} x^{2} + \frac{- 2}{3 !} x^{3} + \frac{9}{4 !} x^{4} + \frac{- 44}{5 !} x^{5} + \dots

Refine

= 1 + \frac{1}{2} x^{2} - \frac{1}{3} x^{3} + \frac{3}{8} x^{4} - \frac{11}{30} x^{5} + \dots

\frac{d y}{d x} + a y + b = 0

x \to 3 - lim (\frac{x + 3}{2})

\int_{0}^{x^{4}} cos (t^{2}) d t

(2, 5), (7, 3)

\int (x^{\frac{1}{2}} + 3 x + 2) d x

What is taylor (e^x)/(1+x) ?
The answer to taylor (e^x)/(1+x) is 1+1/2 x^2-1/3 x^3+3/8 x^4-11/30 x^5+\ldots