InterviewSolution
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InterviewSolution
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Evaluate `lim_(ntooo) (1)/(1+n^(2))+(2)/(2+n^(2))+...+(n)/(n+n^(2)).` |
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Answer» `P_(n)=underset(ntooo)lim(1)/(1+n^(2))+(2)/(2+n^(2))+...+(n)/(n+n^(2))` Now, `P_(n)ltunderset(ntooo)lim(1)/(1+n^(2))+(2)/(1+n^(2))+...+(n)/(1+n^(2))` `=(1)/(1+n^(2))(1+2+3+...+n)` `=(n(n+1))/(2(1+n^(2)))` Also, `P_(n)gt(1)/(n+n^(2))+(2)/(n+n^(2))+(3)/(n+n^(2))+...+(n)/(n+n^(2))` `=(n(+1))/(2(1+n^(2)))` Thus `(n(n+1))/(2(n+n^(2)))ltP_(n)lt(n(n+1))/(2(n+n^(2)))` or`" "underset(ntooo)lim(n(n+1))/(2(n+n^(2)))ltunderset(ntooo)limP_(n)lt(n(n+1))/(2(n+n^(2)))` or` underset(ntooo)lim(1(1+((1))/(n)))/(2((1)/(n)+1))ltunderset(ntooo)limltP_(n)ltunderset(ntooo)lim(1(1+((1))/(n)))/(2((1)/(n^(2))+1))` or `1/2ltunderset(ntooo)limP_(n)lt1/2` or `underset(ntooo)limP_(n)=1/2` |
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