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Evaluate lim(ntooo) ((1^(2)-2^(2)+3^(2)-...

Evaluate `lim_(ntooo) ((1^(2)-2^(2)+3^(2)-4^(2)+5^(2)+...n" terms"))/(n^(2)).`

Text Solution

Verified by Experts

The correct Answer is:
`1//2" or "-1//2`
When n is even:
Given series
`1^(2)-2^(2)+3^(2)-4^(2)+...-n^(2)`
`(1^(2)-2^(2))+(3^(2)-4^(2))+...[(n-1)^(2)-n^(2)]`
`=-(1+2+3+4+...+n)`
`=-(n(n+1))/(2)`
`:. "Given "L=underset(ntooo)lim(n(n+1))/(2n^(2))=-1/2`
When n is odd:
Given series
`1^(2)-2^(2)+3^(2)-4^(2)+...+n^(2)`
`=-1(1+2+3+...+(n-1))+n^(2)`
`=-(n(n-1))/(2)+n^(2)`
`=(n(n+1))/(2)`
`:." Given "L=underset(ntooo)lim(n(n+1))/(2n^(2))=1/2`
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