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Question
The value of limit n tends to infinity ((1.5)n + {(1+0.0001)10000}n)1/n
{} denotes Greatest Integer function is?
Options: A.1 B.1/2 C.Does not exist D.2
The value of limit n tends to infinity ((1.5)n + {(1+0.0001)10000}n)1/n
{} denotes Greatest Integer function is?
Options: A.1 B.1/2 C.Does not exist D.2
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Solution
It is well known that lim n—> inf (1 + 1/n)^n evaluates to e.value of e=2.7,so the greatest integer is {2.7}=2
(1 + 1/10000)^10000=1+100000/100000=1+1=2
((3/2)n+(2)n)1/n take 1/2 outside we get ,((1/2)n)1/n((3/4)n+1)1/n=1/2((3/4)n+1)1/n [(1/2)nx1/n=1/2]
[1+(3/4)n]1/n =[1+(1/n)(3/4)n)1/n [ by binomial theorem, (1+x)n=1+nx ]as n tends to infinity,(1/n)(3/4)n becomes zero
then,1/2((3/4)n+1)1/n=1/2
(1 + 1/10000)^10000=1+100000/100000=1+1=2
((3/2)n+(2)n)1/n take 1/2 outside we get ,((1/2)n)1/n((3/4)n+1)1/n=1/2((3/4)n+1)1/n [(1/2)nx1/n=1/2]
[1+(3/4)n]1/n =[1+(1/n)(3/4)n)1/n [ by binomial theorem, (1+x)n=1+nx ]as n tends to infinity,(1/n)(3/4)n becomes zero
then,1/2((3/4)n+1)1/n=1/2
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