I am trying to solve the following inequality:
$$2+\left(\frac{3}{2}\right)^{\frac{1}{2}}+\left(\frac{4}{3}\right)^{\frac{1}{3}}+\cdots+\left(\frac{n+1}{n}\right)^{\frac{1}{n}}
but I do not know how to do that. Please help me solve this inequality.
I am trying to solve the following inequality:
$$2+\left(\frac{3}{2}\right)^{\frac{1}{2}}+\left(\frac{4}{3}\right)^{\frac{1}{3}}+\cdots+\left(\frac{n+1}{n}\right)^{\frac{1}{n}}
but I do not know how to do that. Please help me solve this inequality.
Use the fact that $(1 + \frac1n)^{\frac1n} < 1 + \frac{1}{n^2}$ for $n > 1$. Thus your sum is $\sum_{k=1}^n (1 + \frac1k)^{\frac1k} < \sum_{k=1}^n (1 + \frac{1}{k^2}) < n + \frac{\pi^2}{6} < n+2$.
Since $\log(1+x)\leq x$ and $e^x\leq\frac{1}{1-x}$ we have: $$(1+1/k)^{1/k}=\exp\left(\frac{1}{k}\,\log(1+1/k)\right)\leq e^{1/k^2}\leq 1+\frac{1}{k^2-1}=1+\frac{1}{2}\left(\frac{1}{k-1}-\frac{1}{k+1}\right),$$ so: $$\sum_{k=2}^{n}(1+1/k)^{1/k} \leq (n-1)+\sum_{k=2}^n \frac{1}{k^2-1}\leq (n-1)+\sum_{k=2}^{+\infty}\frac{1}{k^2-1}=(n-1)+\frac{3}{4}< n.$$