What is taylor 1/(1+1/5 x^3) ?

The answer to taylor 1/(1+1/5 x^3) is 1-1/5 x^3+1/25 x^6-1/125 x^9+\ldots

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taylor 1/(1+1/5 x^3)

taylor

\frac{1}{1 + \frac{1}{5} x ^{3}}

1 - \frac{1}{5} x^{3} + \frac{1}{25} x^{6} - \frac{1}{125} x^{9} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{0}{1 !} x + \frac{0}{2 !} x^{2} + \frac{- \frac{6}{5}}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{\frac{144}{5}}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{- \frac{72576}{25}}{9 !} x^{9} + \frac{0}{10 !} x^{10} + \dots

Refine

= 1 - \frac{1}{5} x^{3} + \frac{1}{25} x^{6} - \frac{1}{125} x^{9} + \dots

What is taylor 1/(1+1/5 x^3) ?
The answer to taylor 1/(1+1/5 x^3) is 1-1/5 x^3+1/25 x^6-1/125 x^9+\ldots