SAJC Sigma Notation Tutorial Q1
By considering $\text{f}\left( r \right)-\text{f}\left( r+1 \right)$ where $\text{f}\left( r \right)=\ln \left( 1+\frac{1}{r} \right)$, find ${{S}_{n}}$, the sum of the first $n$ terms of the series $\ln \left( \frac{{{2}^{2}}}{1\times 3} \right)+\ln \left( \frac{{{3}^{2}}}{2\times 4} \right)+\ln \left( \frac{{{4}^{2}}}{3\times 5} \right)+…$.
Show that ${{S}_{\infty }}$ exists and find its exact value.
Find the smallest value of $n$ for which ${{S}_{n}}$ is within $0.1$ of the sum to infinity.
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