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

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

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

taylor

\frac{( e ^{t} )}{1 + t}

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

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

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

Refine

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

x^{2} + x, at x = 1, x = 4

\frac{d}{d x} (\frac{24}{( x + 1 ) ^{5}})

\frac{1}{x}

y = 2 x, y = 9 x^{2}

\frac{d ^{2}}{d x ^{2}} (\frac{1}{x ^{5}})

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