TITLE: What is $\lim_{n\to\infty} \left(\sum_{k=1}^n \frac1k\right) / \left(\sum_{k=0}^n \frac1{2k+1}\right)$?
QUESTION [5 upvotes]: I have the following problem:

Evaluate
$$ \lim_{n\to\infty}{{1+\frac12+\frac13 +\frac14+\ldots+\frac1n}\over{1+\frac13 +\frac15+\frac17+\ldots+\frac1{2n+1}}} $$

I tried making it into two sums, and tried to make it somehow into an integral, but couldn't find an integral.
The sums I came up with,
$$ \lim_{n\to\infty} { \sum_{k=1}^n {\frac1k} \over {\sum_{k=0}^n {\frac{1}{2k+1}}}} $$

REPLY [2 votes]: $$H_n = \log n + O(1)$$ so
$$H_{2n+1} - \tfrac{1}{2} H_n = \log(2n + 1) - \tfrac{1}{2} \log n + O(1) = \tfrac{1}{2} \log n + O(1)$$
whence
$$\frac{H_n}{H_{2n+1} - \tfrac{1}{2}H_n} = \frac{\log n + O(1)}{\tfrac{1}{2}\log n + O(1)} \to 2.$$