What is taylor e^{(t^2)/2} ?

The answer to taylor e^{(t^2)/2} is 1+1/2 t^2+1/8 t^4+1/48 t^6+1/384 t^8+\ldots

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taylor e^{(t^2)/2}

taylor

e^{\frac{t ^{2}}{2}}

1 + \frac{1}{2} t^{2} + \frac{1}{8} t^{4} + \frac{1}{48} t^{6} + \frac{1}{384} t^{8} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{0}{1 !} t + \frac{1}{2 !} t^{2} + \frac{0}{3 !} t^{3} + \frac{3}{4 !} t^{4} + \frac{0}{5 !} t^{5} + \frac{15}{6 !} t^{6} + \frac{0}{7 !} t^{7} + \frac{105}{8 !} t^{8} + \dots

Refine

= 1 + \frac{1}{2} t^{2} + \frac{1}{8} t^{4} + \frac{1}{48} t^{6} + \frac{1}{384} t^{8} + \dots

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

\int \frac{csc ^{2} ( 2 x )}{4 - cot ( 2 x )} d x

\int_{- 1}^{1} x (1 - x^{2}) d x

\int sin^{2} (x) cos (n x) d x

\frac{d}{d x} (\frac{cos ( x )}{2 x})

What is taylor e^{(t^2)/2} ?
The answer to taylor e^{(t^2)/2} is 1+1/2 t^2+1/8 t^4+1/48 t^6+1/384 t^8+\ldots