What is taylor e^{2x},x=0 ?

The answer to taylor e^{2x},x=0 is 1+2x+2x^2+4/3 x^3+2/3 x^4+\ldots

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taylor e^{2x},x=0

taylor

e^{2 x}, x = 0

1 + 2 x + 2 x^{2} + \frac{4}{3} x^{3} + \frac{2}{3} x^{4} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{2}{1 !} x + \frac{4}{2 !} x^{2} + \frac{8}{3 !} x^{3} + \frac{16}{4 !} x^{4} + \dots

Refine

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

x \to 2 lim (- 4 x^{2} - 3 x + 6)

\int 6 t d t

\frac{\partial}{\partial y} (5 y - 2 x)

2 y^{^{''}} + 8 y^{^{'}} + 80 y = 100 cos (8 t)

\int x^{2} + x - 6 d x

What is taylor e^{2x},x=0 ?
The answer to taylor e^{2x},x=0 is 1+2x+2x^2+4/3 x^3+2/3 x^4+\ldots