I have this inequality that I don't know how to prove. Possibly the inequality between means might be useful. For $n \in \mathbb{N}$:
$$\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1} \ge 1$$
I have this inequality that I don't know how to prove. Possibly the inequality between means might be useful. For $n \in \mathbb{N}$:
$$\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1} \ge 1$$
By C-S we obtain: $$\sum_{k=1}^{2n+1}\frac{1}{n+k}=\sum_{k=0}^{n-1}\left(\frac{1}{n+1+k}+\frac{1}{3n+1-k}\right)+\frac{1}{2n+1}\geq$$ $$\geq\sum_{k=0}^{n-1}\frac{(1+1)^2}{n+1+k+3n+1-k}+\frac{1}{2n+1}=\sum_{k=0}^{n-1}\frac{2}{2n+1}+\frac{1}{2n+1}=1.$$ Done!
Write this as $$ \sum_{j=1}^{2n+1} \dfrac{1}{n+j} \ge \int_1^{2n+2} \dfrac{dx}{n+x} = \ln \left(\frac{3n+2}{n+1}\right)$$
Now $\dfrac{3n+2}{n+1} \ge e$ for $n > \dfrac{e-2}{3-e} \approx 2.549$. Do the cases $n=1$ and $2$ separately.
We induct on $n$. For $n=1$, it's trivial. Now assume it holds for $n=k$, then $$ \frac{1}{(k+1)+1}+\cdots+\frac{1}{3(k+1)}+\frac{1}{3(k+1)+1}=\frac{1}{k+2}+\cdots+\frac{1}{3k+4} $$ $$ =(\frac{1}{k+1}+\cdots+\frac{1}{3k+1})-\frac{1}{k+1}+\frac{1}{3k+2}+\frac{1}{3k+3}+\frac{1}{3k+4} $$ Since the first expression is at least $1$ by the inductive hypothesis, it suffices to show $$ \frac{1}{3k+2}+\frac{1}{3k+3}+\frac{1}{3k+4}\ge \frac{1}{k+1} $$ But by Cauchy-Schwarz we obtain $$ (3k+2+3k+3+3k+4)(\frac{1}{3k+2}+\frac{1}{3k+3}+\frac{1}{3k+4})\ge9 $$ Which is equivalent to what we want after dividing by $9(k+1)$. So the inductive hypothesis always holds.
We can also use user8268's hint and apply Cauchy or AM-HM directly: $$ (k+1+\cdots+3k+1)(\frac{1}{k+1}+\cdots+\frac{1}{3k+1})\ge (2k+1)^2 $$ So it suffices to show $$ (2k+1)^2\ge (k+1+\cdots+3k+1)=(1+\cdots+3k+1)-(1+\cdots+k) $$ or $$ 4k^2+4k+1\ge \frac{(3k+1)(3k+2)}{2}-\frac{k(k+1)}{2}=\frac{8k^2+8k+2}{2}=4k^2+4k+1 $$ but this is clearly true.
In general, if $f$ is a convex function and $p_j$ are non-negative coefficients such that $\sum p_j = 1$ then
$$ \sum_j p_j \, f(x_j) \geq f(\sum_j p_j \, x_j). $$
(This is Jensen's inequality.) Take $f(x) = 1/x$, $p_j = 1/(2n + 1)$ and $x_j = n + j$ for $j \in \{1, \dotsc, 2n+1 \}$ and your inequality follows.
It follows immediately from arithmetic mean harmonic mean inequality. Recall that given a set of positive numbers, $\{a_k\}_{k=1}^{k=m}$, we have that $$\dfrac{\displaystyle\sum_{k=1}^{m} a_k}m \geq \dfrac{m}{\displaystyle \sum_{k=1}^m \dfrac1{a_k}}$$ $$\dfrac{\displaystyle \sum_{k=1}^{2n+1} \dfrac1{n+k}}{2n+1} \geq \dfrac{2n+1}{\displaystyle \sum_{k=1}^{2n+1} (n+k)}$$ $$\displaystyle \sum_{k=1}^{2n+1} (n+k) = n(2n+1) + \dfrac{(2n+1)(2n+2)}2 = (2n+1)^2$$ Hence, $$\dfrac{\displaystyle \sum_{k=1}^{2n+1} \dfrac1{n+k}}{2n+1} \geq \dfrac{2n+1}{(2n+1)^2}$$ Hence, $$\displaystyle \sum_{k=1}^{2n+1} \dfrac1{n+k} \geq 1$$