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Question Number 11838 by Peter last updated on 02/Apr/17

\frac{2^{2} + 1}{2^{2} - 1} + \frac{3^{2} + 1}{3^{2} - 1} + \frac{4^{2} + 1}{4^{2} - 1} + \dots . + \frac{20^{2} + 1}{20^{2} - 1} = \dots . ?

Answered by ajfour last updated on 02/Apr/17

T_{r} = \frac{r^{2} + 1}{r^{2} - 1} = \frac{r^{2} - 1 + 2}{r^{2} - 1}

= 1 + \frac{2}{(r - 1) (r + 1)}

T_{r} = 1 + \frac{1}{(r - 1)} - \frac{1}{(r + 1)}

S = \underset{r = 2}{\sum^{r = 20}} T_{r}

= (1 + \frac{1}{1} - \frac{1}{3})

+ (1 + \frac{1}{2} - \frac{1}{4})

+ (1 + \frac{1}{3} - \frac{1}{5})

+ (1 + \frac{1}{4} - \frac{1}{6})

+ \dots .

\dots + (1 + \frac{1}{17} - \frac{1}{19})

+ (1 + \frac{1}{18} - \frac{1}{20})

+ (1 + \frac{1}{19} - \frac{1}{21})

S = 19 + 1 + \frac{1}{2} - \frac{1}{20} - \frac{1}{21}

S = 20 + \frac{169}{420} .

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