By considering $\mathrm{f}\left( r \right)-\mathrm{f}\left( r+1 \right)$ where $\mathrm{f}\left( r \

Question

By considering $\mathrm{f}\left( r ight)-\mathrm{f}\left( r+1 ight)$ where $\mathrm{f}\left( r ight)=\ln \left( 1+\frac{1}{r} ight)$, find ${{S}_{n}}$, the sum of the first $n$ terms of the series $\ln \left( \frac{{{2}^{2}}}{1	imes 3} ight)+\ln \left( \frac{{{3}^{2}}}{2	imes 4} ight)+\ln \left( \frac{{{4}^{2}}}{3	imes 5} ight)+...$. 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.

Accepted Answer

$S_n = \ln 2 - \ln\left(1 + \dfrac{1}{n+1}\right)$, $S_{\infty} = \ln 2$, smallest $n = 9$

SAJC Sigma Notation Tutorial Q1

SAJC Sigma Notation Tutorial Q1

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