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Prove that loge ((n^2)/(n^2-1))=1/(n^2)...

Prove that `log_e ((n^2)/(n^2-1))=1/(n^2)+1/(2n^4)+1/(3n^6)+........oo`

Text Solution

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L.H.S. ` log_(n) ((n^(2))/(n^(2) -1)) =- log_(e) ((n^(2) -1)/(n^(2))) `
` = - log_(e) (1-(1)/(n^(2))) " "- lt (1)/(n^(2) )lt 1 `
` [ - (1)/(n^(2)) - (1)/(2n^(4)) - (1)/(3n^(6)) - ...oo]`
`= (1)/(n^(2)) + (1)/(2.n^(4))+(1)/ (3.n^(6)) + ...oo` .
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