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Sum of the series (x+y)(x-y) +1/(2!)(x...

Sum of the series
`(x+y)(x-y) +1/(2!)(x+y) (x-y) (x^(2)+y^(2))`
` 1/(3!)(x+y)(x-y)(x^(4)+y^(4)+x^(2)y^(2))+...` is

A

`e^(x)+e^(y)`

B

`e^(x-e^(y)`

C

`e^(x^(2))+e^(y^(2))`

D

` e^(x^(2))-e^(y^(2))`

Text Solution

Verified by Experts

The correct Answer is:
c
`(x+y)(x-y)+1/(2!)(x+y)(x-y) (x^(2)+y^(2))`
` +1/(3!)(x+y) (x-y)(x^(4)+y^(4)+x^(2)y^(2))+...`
`= (x^(2)-y^(2)) +1/(2!)(x^(4)-y^(4))+1/(3!)(x^(6)-y^(6))+...`
` = (x^(2) -y^(2))+1/(2!)(x^(4)-y^(4))+1/(3!)(x^(6)-y^(6))+...`
` = ( x^(2) + 1/(2!) x^(4) +(x^(6))/(3!) +...)-(y^(2)+(y^(4))/(2!) +(y^(6))/(3!)+...)` ,
` = (1+(x^(2))/(1!)+(x^(2))^(2)/(2!)+(x^(2))^(3)/(3!)+...)`
` -(1+(y^(2))/(1!) +(y^(2))^(2)/(2!)+(y^(2))^(3)/(3!)+...)`
` = e^(x^(2)) -e^(y^(2))`
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