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How to find the first three terms of the Taylor series around $b=c$ of

$$ \int^{\frac{1}{b}}_{a}f(b,x) dx $$

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1 Answer 1

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Assuming you mean the taylor series of the function $F$ where $$ F(b)=\int^{\frac{1}{b}}_{a}f(b,x) dx $$ Let's find the coefficients of the Taylor Series.

We know by Liebniz Rule that, assuming $f$ has a continuous partial derivative with respect to $b$ (I believe this is necessary and sufficient, please correct me if I am wrong), $$ F'(b)=\int^{\frac{1}{b}}_{a}\frac{\partial f}{\partial b}(b,x) dx-\frac{1}{b^2}f(b,1/b) $$ And similarly for the second and third order derivatives. Then plug in $c$ for your coefficients.

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  • $\begingroup$ the derivative of the function I defined in the first line with respect to $b$ $\endgroup$ Sep 13, 2016 at 14:26
  • $\begingroup$ What does $F'(b)$ represent here? Do you mean that $F(c)$ represent the first term and $b\, F'(c)$ represent the second term in the Taylor series? $\endgroup$
    – gbd
    Sep 13, 2016 at 14:27
  • $\begingroup$ the coefficients on the first and second term respectively, yes $\endgroup$ Sep 13, 2016 at 14:28

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