Sunday, 3 September 2017

real analysis - Evaluate limntoinfty((15)n+([(1+0.0001)10000])n)frac1n



Evaluate lim Here [.] denotes the greatest integer function.



My Try : I know how to solve this kind of problem :\lim_{n \to \infty} ((a)^n +(b)^n)^{\frac{1}{n}} where a, b \geq 0. But here I can not find ([(1+0.0001)^{10000}])?



Can anyone please help me out?



Thank You.


Answer




Since for all n\in \mathbb{N}



2\le\left(1+\frac{1}{n}\right)^n \le e < 3



we have that



\left((15)^n +\left[\left(1+\frac1{10000}\right)^{10000}\right]^n\right)^{\frac{1}{n}}= \left((15)^n +2^n\right)^{\frac{1}{n}}=15 \left(1 +(2/15)^n\right)^{\frac{1}{n}}\to 15


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