One Variable Calculus

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Calculus of One Variable

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$\displaystyle \begin{aligned} a_n:=\ln\left(\prod_{k=0}^n \binom{n}{k}\right)=\sum_{k=0}^n (\ln n! - \ln k! - \ln(n-k)!)=(n+1)\ln n! - 2\sum_{k=0}^n \ln k! \\ =(n+1)\sum_{i=1}^n \ln i - 2\sum_{k=1}^n\sum_{i=1}^k \ln i =(n+1)\sum_{i=1}^n \ln i - 2 \sum_{i=1}^n\sum_{k=i}^n\ln i \\ =(n+1)\sum_{i=1}^n \ln i - 2\sum_{i=1}^n(n-i+1)\ln i=\sum_{i=1}^n(2i-n-1)\ln i \\ =\sum_{i=1}^n (2i-n)\ln \left(\frac{i}{n}\right)+\sum_{i=1}^n(2i-n)\ln n - \sum_{i=1}^n \ln i \\ =\sum_{i=1}^n (2i-n)\ln \left(\frac{i}{n}\right)+n\ln n- \ln n!. \end{aligned}$

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$\displaystyle J_k:=\int_0^{\frac{\pi}{2}}x^k \cos(2nx) \ dx,$

One-Variable Calculus – Calculus + epsilon

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One Variable Calculus

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