Sum of series of fractions I am trying to find the $f$ formula that returns the sum of the series created by fractions that have constant nominator and shifting by one denominator. 
Here are some examples:
$$f(3) = \frac{3}{1} + \frac{3}{2} + \frac{3}{3} = 5.5$$
or
$$f(4) = \frac{4}{1} + \frac{4}{2} + \frac{4}{3} + \frac{4}{4} = 8.33$$
or
$$f(5) = \frac{5}{1} + \frac{5}{2} + \frac{5}{3} + \frac{5}{4} + \frac{5}{5} = 11.4166$$
or
$$f(200) = \frac{200}{1} + \frac{200}{2} + \frac{200}{3} + ... + \frac{200}{200} = 1175.6062$$
Does anyone have an idea on how to calculate the sum of this series?

The numbers
$$1+\frac12+\frac13+\cdots+\frac1n$$
are called the harmonic numbers and often denoted $H_n$. There is
no simple closed formula for $H_n$, but $H_n$ is approximately $\ln n+\gamma$ for large $n$, where $\gamma$ is Euler's constant.
You are considering $nH_n$.
