start{align*}

sum_{n=1}^{infty} frac{1}{n(n+1)(2n+1)}&=sum_{n=1}^{infty}left(frac{1}{n+1}-frac{4}{2n+1}+frac{1}{n}proper)

&=sum_{n=1}^{infty} int_0^1 x^{n}-4x^{2n}+x^{n-1} {rm d}x

&=int_0^1 sum_{n=1}^{infty} x^{n}-4x^{2n}+x^{n-1} {rm d}x

&=int_0^1 left(frac{x}{1-x}+frac{4x^2}{x^2-1}+frac{1}{1-x} proper) {rm d}x

&=int_0^1frac{3x+1}{x+1} {rm d}x

&=3x-2ln(x+1)bigg|_0^1

&=3-2ln 2

finish{align*}

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