I want my book to be away from this table Story

So I have a book that I want to separate from my table with nothing but other books. I want to know how many books do I need to achieve this with \$n\$ book lengths.

Here's a visualization that my friend at Wolfram drew for me:



More information about the topic in Wolfram and Wikipedia.

Challenge

Given an integer input \$n\$, output how many books needed for the top book to be \$n\$ book length away from the table horizontally.
or
Find the smallest integer value of \$m\$ for input \$n\$ in the following inequality.
$$\sum_{i=1}^{m}\frac{1}{2i} \geq n$$

Edit: for fractions use at least a IEEE single-precision floating point. sorry for editing challenge after posting

(OEIS A014537)

Test cases

 1          4
 2         31
 3        227
 5      12367
10  272400600
Octave, 41 40 33 bytes
1 byte saved thanks to @Dennis

@(n)find(cumsum(.5./(1:9^n))&gt;n,1)

Try it online!
Explanation
This uses the fact that harmonic numbers can be lower-bounded by a logarithmic function.
Also, the &gt;= comparison can be replaced by &gt; because harmonic numbers cannot be even integers (thanks, @Dennis!).
@(n)                                   % Anonymous function of n
                     1:9^n             % Range [1 2 ... 9^n]
                .5./(     )            % Divide .5 by each entry
         cumsum(           )           % Cumulative sum
                            &gt;n         % Is each entry greater than n?
    find(                     ,1)      % Index of first true entry