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

((2^2 +1)/(2^2 −1)) + ((3^2 +1)/(3^2 −1)) + ((4^2 +1)/(4^2 −1)) + .... + ((20^2 +1)/(20^2 −1)) = ....?

$$\frac{\mathrm{2}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}^{\mathrm{2}} −\mathrm{1}}\:+\:\frac{\mathrm{3}^{\mathrm{2}} +\mathrm{1}}{\mathrm{3}^{\mathrm{2}} −\mathrm{1}}\:+\:\frac{\mathrm{4}^{\mathrm{2}} +\mathrm{1}}{\mathrm{4}^{\mathrm{2}} −\mathrm{1}}\:+\:....\:+\:\frac{\mathrm{20}^{\mathrm{2}} +\mathrm{1}}{\mathrm{20}^{\mathrm{2}} −\mathrm{1}}\:=\:....? \\ $$

Answered by ajfour last updated on 02/Apr/17

T_r = ((r^2 +1)/(r^2 −1)) = ((r^2 −1+2)/(r^2 −1))    =1+(2/((r−1)(r+1)))    T_r  = 1+(1/((r−1)))−(1/((r+1)))  S= Σ_(r=2) ^(r=20) T_r       = (1+(1/1)−(1/3))        +( 1+(1/2)−(1/4))        +(1+(1/3)−(1/5))        + (1+(1/4)−(1/6))         +....     ...+(1+(1/(17))−(1/(19)))         +(1+(1/(18))−(1/(20)))         +(1+(1/(19))−(1/(21)))  S = 19+1+(1/2)−(1/(20))−(1/(21))  S= 20+((169)/(420))  .

$${T}_{{r}} =\:\frac{{r}^{\mathrm{2}} +\mathrm{1}}{{r}^{\mathrm{2}} −\mathrm{1}}\:=\:\frac{{r}^{\mathrm{2}} −\mathrm{1}+\mathrm{2}}{{r}^{\mathrm{2}} −\mathrm{1}} \\ $$$$\:\:=\mathrm{1}+\frac{\mathrm{2}}{\left({r}−\mathrm{1}\right)\left({r}+\mathrm{1}\right)}\: \\ $$$$\:{T}_{{r}} \:=\:\mathrm{1}+\frac{\mathrm{1}}{\left({r}−\mathrm{1}\right)}−\frac{\mathrm{1}}{\left({r}+\mathrm{1}\right)} \\ $$$${S}=\:\underset{{r}=\mathrm{2}} {\overset{{r}=\mathrm{20}} {\sum}}{T}_{{r}} \\ $$$$\:\:\:\:=\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{3}}\right) \\ $$$$\:\:\:\:\:\:+\left(\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{4}}\right) \\ $$$$\:\:\:\:\:\:+\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{5}}\right) \\ $$$$\:\:\:\:\:\:+\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{6}}\right) \\ $$$$\:\:\:\:\:\:\:+.... \\ $$$$\:\:\:...+\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{17}}−\frac{\mathrm{1}}{\mathrm{19}}\right) \\ $$$$\:\:\:\:\:\:\:+\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{18}}−\frac{\mathrm{1}}{\mathrm{20}}\right) \\ $$$$\:\:\:\:\:\:\:+\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{19}}−\frac{\mathrm{1}}{\mathrm{21}}\right) \\ $$$${S}\:=\:\mathrm{19}+\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{20}}−\frac{\mathrm{1}}{\mathrm{21}} \\ $$$${S}=\:\mathrm{20}+\frac{\mathrm{169}}{\mathrm{420}}\:\:. \\ $$

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