Evaluation of $$\lim_{n \rightarrow \infty}\bigg[\frac{1}{n}+\frac{1}{n+2}+\frac{1}{n+4}+\cdots \cdots +\frac{1}{3n}\bigg]$$
My work: Using Riemann sum
$$\lim_{n\rightarrow \infty}\sum^{n}_{r=0}\frac{1}{n+2r}=\lim_{n\rightarrow\infty}\sum^{n}_{r=0}\frac{1}{1+2\frac{r}{n}}\cdot \frac{1}{n}$$
Put $\displaystyle \frac{r}{n}=x$ and $\displaystyle \frac{1}{n}=dx$ and changing limits
$$\int^{1}_{0}\frac{1}{1+2x}dx=\frac{1}{2}\ln|1+2x|\bigg|^{1}_{0}=\frac{1}{2}\ln(3)$$
But would it be possible to do the problem without usage of the Rienmann sum? as in, could one solve such infinite sums in a method alternate to converting the sum into an integral.
Best Answer
For even $n=2m$ we have \begin{align} \sum_{r=0}^{2m}\frac{1}{2m+2r} &=\frac{1}{2}\sum_{r=0}^{2m}\frac{1}{m+r}=\\ &=\frac{1}{2}\left(\sum_{r=1}^{3m}\frac{1}{r}-\sum_{r=1}^{m-1}\frac{1}{r}\right)=\frac{1}{2}(H_{3m}-H_{m-1}), \end{align} where $$ H_n=\sum_{r=1}^n\frac{1}{r} $$ are the Harmonic numbers. Given the known relation $$ \lim_{n\to\infty}(H_n-\log n)=\gamma $$ we have \begin{align} &\lim_{m\to\infty}\sum_{r=0}^{2m}\frac{1}{2m+2r} =\frac{1}{2}\lim_{m\to\infty}(H_{3m}-H_{m-1})=\\ &\qquad=\frac{1}{2}\lim_{m\to\infty}[(H_{3m}-\log(3m))+\log(3m)-(H_{m-1}-\log(m-1))-\log(m-1)]=\\ &\qquad=\frac{1}{2}\lim_{m\to\infty}[\gamma+\log(3m)-\gamma-\log(m-1)]=\\ &\qquad=\frac{1}{2}\lim_{m\to\infty}\log\left(\frac{3m}{m-1}\right)=\frac{1}{2}\log 3. \end{align}
For odd $n=2m+1$, taking into account \begin{align} &\frac{1}{n}+\frac{1}{n+2}+\ldots+\frac{1}{3n-2}+\frac{1}{3n}=\\ &=\left(\frac{1}{n}+\frac{1}{n+1}+\ldots+\frac{1}{3n-1}+\frac{1}{3n}\right)-\left(\frac{1}{n+1}+\frac{1}{n+3}+\ldots+\frac{1}{3n-3}+\frac{1}{3n-1}\right) \end{align} we can write \begin{align} \sum_{r=0}^{2m+1}\frac{1}{2m+1+2r} &= \sum_{s=0}^{4m+2}\frac{1}{2m+1+s}-\sum_{r=0}^{2m}\frac{1}{2m+2+2r}=\\ &= \sum_{s=0}^{4m+2}\frac{1}{2m+1+s}-\frac{1}{2}\sum_{r=0}^{2m}\frac{1}{m+1+r}=\\ &= H_{6m+3}-H_{2m}-\frac{1}{2}[H_{3m+1}-H_{m}] \end{align} and \begin{align} \lim_{m\to\infty}\sum_{r=0}^{2m+1}\frac{1}{2m+1+2r} &= \lim_{m\to\infty}\left(H_{6m+3}-H_{2m}-\frac{1}{2}[H_{3m+1}-H_{m}]\right)=\\ &= \lim_{m\to\infty}\left(\log(6m+3)-\log(2m)-\frac{1}{2}[\log(3m+1)-\log(m)]\right)=\\ &= \lim_{m\to\infty}\left(\log\left(\frac{6m+3}{2m}\right)-\frac{1}{2}\log\left(\frac{3m+1}{m}\right)\right)=\frac{1}{2}\log 3 \end{align}
Alternative proof
Let's rewrite the sum as $$ \frac{1}{2}\sum_{r=0}^n\frac{1}{\frac{n}{2}+r}=\frac{1}{2}\left[\psi\left(\frac{3n+2}{2}\right)-\psi\left(\frac{n}{2}\right)\right], $$ where $\psi$ is the digamma function and where we used the difference equation $$ \psi(x+N)-\psi(x)=\sum_{k=0}^{N-1}\frac{1}{x+k}, $$ see Digamma::Recurrence formula and characterization.
Now, taking into account the following inequality, valid for $x>0$ $$ \log x-\frac{1}{x}\leq\psi(x)\leq\log x-\frac{1}{2x}, $$ see Digamma::Inequalities, we have $$ \log\left(\frac{3n+2}{n}\right)-\frac{2}{3n+2}+\frac{1}{n}\leq\psi\left(\frac{3n+2}{2}\right)-\psi\left(\frac{n}{2}\right)\leq \log\left(\frac{3n+2}{n}\right)-\frac{1}{3n+2}+\frac{2}{n} $$ and by the squeeze theorem, we get the result.