What is taylor e^{5x},0 ?

The answer to taylor e^{5x},0 is 1+5x+25/2 x^2+125/6 x^3+625/24 x^4+\ldots

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taylor e^{5x},0

taylor

e^{5 x}, 0

1 + 5 x + \frac{25}{2} x^{2} + \frac{125}{6} x^{3} + \frac{625}{24} x^{4} + \dots

Solution steps

Apply the Taylor Formula

= 1 + \frac{\frac{d}{d x} ( e ^{5 x} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( e ^{5 x} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( e ^{5 x} ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= 1 + \frac{5}{1 !} x + \frac{25}{2 !} x^{2} + \frac{125}{3 !} x^{3} + \frac{625}{4 !} x^{4} + \dots

Refine

= 1 + 5 x + \frac{25}{2} x^{2} + \frac{125}{6} x^{3} + \frac{625}{24} x^{4} + \dots

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What is taylor e^{5x},0 ?
The answer to taylor e^{5x},0 is 1+5x+25/2 x^2+125/6 x^3+625/24 x^4+\ldots