Induction with nth root of n

Question

I am trying to prove by induction that $\sqrt[n]{n}<2-\frac{1}{n}$ where $n\ge2$. It seemed simple at first, but I am stuck with $log(2n-1)$ in the RHS. I am in an elementary undergraduate Maths course. Please help me out.

Answer 1

By Bernoulli $$\left(2-\frac{1}{n}\right)^n=\left(1+\left(1-\frac{1}{n}\right)\right)^n>1+n\left(1-\frac{1}{n}\right)=n.$$

Answer 2

You need to show , $$n\leq \left(2-\frac{1}{n}\right)^n$$ It holds for the base case $n=2$.

Assuming it holds for $n=k$,

$$k<\left(2-\frac{1}{k}\right)^k$$

we need to show the same for $n=k+1$ i.e.,

$$k+1<\left(2-\frac{1}{k+1}\right)^{k+1}$$

let's try to prove something stronger,

$$k+1<\left(2-\frac{1}{k}\right)^{k+1}<\left(2-\frac{1}{k+1}\right)^{k+1}$$

using the the relation for $n=k$,

$$\left(2-\frac{1}{k}\right)^{k+1}>\left(2-\frac{1}{k}\right)\times k=2k-1>k+1 \qquad\forall k>2$$