taylor e^t

Solution

taylor

e^{t}

n = 0 \sum \infty \frac{x ^{n}}{n !}

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{1}{1 !} t + \frac{1}{2 !} t^{2} + \frac{1}{3 !} t^{3} + \frac{1}{4 !} t^{4} + \dots

The series sum representation

= n = 0 \sum \infty \frac{x ^{n}}{n !}

\frac{d}{d x} (\frac{250000 x ^{2}}{( 2 x + 1 ) ^{2}})

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\int 10 tan^{3} (x) d x

What is taylor e^t ?
The answer to taylor e^t is sum from n=0 to infinity of (x^n)/(n!)