Let \( g(x)=\log f(x) \), where \( f(x) \) is twice differentiable positive function on \( (0, \....

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Let \( g(x)=\log f(x) \), where \( f(x) \) is twice differentiable
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positive function on \( (0, \infty) \) such that \( f(x+1)=x f(x) \).
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Then, for \( N=1,2,3, \ldots g^{\prime \prime}\left(N+\frac{1}{2}\right)-g^{\prime \prime}\left(\frac{1}{2}\right)= \)
(1) \( -4\left\{1+\frac{1}{9}+\frac{1}{25}+\ldots+\frac{1}{(2 N-1)^{2}}\right\} \)
(2) \( 4\left\{1+\frac{1}{9}+\frac{1}{25}+\ldots+\frac{1}{(2 N-1)^{2}}\right\} \)
(3) \( -4\left\{1+\frac{1}{9}+\frac{1}{25}+\ldots+\frac{1}{(2 N+1)^{2}}\right\} \)
(4) \( 4\left\{1+\frac{1}{9}+\frac{1}{25}+\ldots+\frac{1}{(2 N+1)^{2}}\right\} \)


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