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The series expansion of log(e) [(1 + x...

The series expansion of ` log_(e) [(1 + x^((1 + x))(1-x)^(1-x)]` is
(1)`2[(x^(2))/(1.2) + (x^(4))/(3.4)+(x^(6))/(5.6)+...]`
(2) `[(x^(2))/(1.2) + (x^(4))/(3.4)+(x^(6))/(5.6)+...]`
(3)`2[(x^(2))/(1.2) + (x^(4))/(2.3)+(x^(6))/(3.4)+...]`
(4)`2[(x^(2))/(1.2) -(x^(4))/(2.3)+(x^(6))/(3.4)-...]`

Text Solution

Verified by Experts

L.H.S = `(1 + x) log_(e) (1+ x) + (1 - x) log_(e) (1 - x)` ltbrge` log_(e) (1 + x)(1 -x) + xlog_(e) ((1 + x)/(1 - x))`
= ` - 2[ (x^(2))/(2) + (x^(4))/(4) + (x^(6))/(6) + ...oo] + 2 xx[ x + (x^(3))/(3) + (x^(5))/(5) + ...oo]`
`2[ x^(2) (1 - (1)/(2)) + x^(4) ((1)/(3) - (1)/(4)) + x^(6) ((1)/(5) - (1)/(6)) + ...oo]` .
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