Calculating $\lim _{n\to \infty \:}\left(\frac{1+\frac{1}{2}+\frac{1}{3}+...\frac{1}{n}}{1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{2n+1}}\right)$? https://ift.tt/eA8V8J

Hello everyone how can I calculate the limit of:

$\lim _{n\to \infty \:}\left(\frac{1+\frac{1}{2}+\frac{1}{3}+...\frac{1}{n}}{1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{2n+1}}\right)?$

I tried to convert this to something that looks like Riemann sum $$\lim _{n\to \infty \:}\left(\frac{\sum^n_{k=0}(\frac{1}{k})}{1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{2n+1}}\right)$$

But I don't know how to continue.

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