Evaluate $\lim_{n \to \infty} ((15)^n +([(1+0.0001)^{10000}])^n)^{\frac{1}{n}}$

Question

Evaluate $\lim_{n \to \infty} ((15)^n +([(1+0.0001)^{10000}])^n)^{\frac{1}{n}}$ 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 1

Since for all $n\in \mathbb{N}$

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

we have that

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

Answer 2

Hint: since $$ \lim_n \left(1 + \frac 1n\right)^n = \mathrm e \in [2,3], $$ so the integral part of your expression should be…?

Answer 3

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

Answer 4

Let $a = 15, b = [(1+10^{-4})^{10000}]=3$. We have that a > b. So: $$ \begin{align} (a^n+b^n)^{\frac{1}{n}} &= \exp\left(\frac{1}{n}\ln(a^n+b^n)\right) \\ &= \exp\left(\frac{1}{n}\left(\ln(a^n)+\ln\left(1+\left(\frac{b}{a}\right)^n\right)\right)\right) \\ &= a\cdot\exp\left(\frac{1}{n}\left(\frac{b}{a}\right)^n+o\left(\frac{1}{n}\left(\frac{b}{a}\right)^n\right)\right) \\ & \underset{n\infty}{\longrightarrow}a \end{align} $$.