Friday, October 7, 2016

calculus - How to show limntoinftyan=frac53cdotn2left(n+1right)2?




lim



I am trying to solve it using the squeeze theorem.
I have opened the expression to a_n=\dfrac{5^3\cdot 5^n}{2^{n^2}\cdot2^{2n}\cdot2)}
I think that the LHS should be a_n=\dfrac{2^3\cdot 2^n}{2^{n^2}\cdot2^{2n}\cdot2)}
But as for the RHS I do not find a bigger expression, any ideas?


Answer



Hint: 0<\dfrac{125^n}{2^{(n+1)^2}} < \dfrac{125^n}{2^{n^2}} < \left(\dfrac{1}{2}\right)^n



since \dfrac{125}{2^n} < \dfrac{1}{2} when n > 8.



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