If `"log"_(e) (1)/(1 + x + x^(2) + x^(3))` be expemnded in asecending power of x , show that the coefficient of ` x^(n)`
is `- (1)/(n)` if n is odd or of the form 4m + 2 and `(3)/(2)`if n is of the form 4m

Explanation

`"log"_(e) (1)/(1 + x + x^(2) + x^(3)) = "log" _(e) (1)/((1 + x) (1 + x^(2)))`
` = - log_(e) {(1 + x)(1 + x^(2))}`
` - log_(e) (1 + x) - log (1 + x^(2))`
`- { x - (1)/(2) x^(2) + (1)/(3) x^(3) - ...} - { x^(2) - (1)/(2) x^(4) + (1)/(3) x^(6) - ...}`
Let n be odd . Then there is no term containing ` x^(n)` in the second series .
` therefore ` Coefficient ` x^(n) = - (1)/(n) `
Let n = 4m + 2 . Then coefficient of ` x^(n)` = coefficient of ` x^(4m +2)`
`= (1)/(4m +2) - (1)/(2m +1) = - (1)/(4m +2) = - (1)/(n) `
Let n = 4m. Them coefficient of `x^(n)` coefficient of ` x^(4m)`
` = (1)/(4m) + (1)/(2m) = 3 (1)/(4m) = (3)/(n)`
Hence proved .