What is taylor ln(x),2 ?

The answer to taylor ln(x),2 is ln(2)+1/2 (x-2)-1/8 (x-2)^2+1/24 (x-2)^3-1/64 (x-2)^4+\ldots

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taylor ln(x),2

taylor

ln (x), 2

ln (2) + \frac{1}{2} (x - 2) - \frac{1}{8} (x - 2)^{2} + \frac{1}{24} (x - 2)^{3} - \frac{1}{64} (x - 2)^{4} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

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

Refine

= ln (2) + \frac{1}{2} (x - 2) - \frac{1}{8} (x - 2)^{2} + \frac{1}{24} (x - 2)^{3} - \frac{1}{64} (x - 2)^{4} + \dots

y = 3 x^{\frac{3}{8}}

\frac{d y}{d x} = (1 + y) (1 - 2 x)

\int \frac{9 x}{( x + 3 ) ^{2}} d x

\int (\frac{2}{x ^{5}}) d x

\int \frac{4 x - 1}{x ^{2} - 25} d x

What is taylor ln(x),2 ?
The answer to taylor ln(x),2 is ln(2)+1/2 (x-2)-1/8 (x-2)^2+1/24 (x-2)^3-1/64 (x-2)^4+\ldots