What is taylor x^{-1/2} ?

The answer to taylor x^{-1/2} is 1-1/2 (x-1)+3/8 (x-1)^2-5/16 (x-1)^3+35/128 (x-1)^4+\ldots

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taylor x^{-1/2}

taylor

x^{- \frac{1}{2}}

1 - \frac{1}{2} (x - 1) + \frac{3}{8} (x - 1)^{2} - \frac{5}{16} (x - 1)^{3} + \frac{35}{128} (x - 1)^{4} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{- \frac{1}{2}}{1 !} (x - 1) + \frac{\frac{3}{4}}{2 !} (x - 1)^{2} + \frac{- \frac{15}{8}}{3 !} (x - 1)^{3} + \frac{\frac{105}{16}}{4 !} (x - 1)^{4} + \dots

Refine

= 1 - \frac{1}{2} (x - 1) + \frac{3}{8} (x - 1)^{2} - \frac{5}{16} (x - 1)^{3} + \frac{35}{128} (x - 1)^{4} + \dots

n \to \infty lim (\frac{3 n ^{0.4 + 1} + 7 n}{3 n ^{0.4} + 7 n})

y^{^{''}} + 5 y^{^{'}} = e^{- t}, y (0) = - 0, y^{^{'}} (0) = 1

\int \frac{- y}{cot ( x )} d x

y^{^{''}} + 2 y^{^{'}} - 24 y = 0

x y + y^{^{'}} = 76 x

What is taylor x^{-1/2} ?
The answer to taylor x^{-1/2} is 1-1/2 (x-1)+3/8 (x-1)^2-5/16 (x-1)^3+35/128 (x-1)^4+\ldots