Finding the sum of this series: $1+\frac 12 + \frac 13 + \cdots + \frac 1{50}$ I need to find the sum of this series: $1+\frac 12 + \frac 13 + \cdots + \frac 1{50}$


Please help me find the sum of this series.
There is no nice closed formula for partial harmonic sums. However, there is a technique called partial summation used to estimate these types of sums. Partial summation, or Abel summation, is just integration by parts for the Riemann-Stieltjes integral.

You can look up how to approximate sums by the method of partial summation if you are interested. The estimate it provides for partial harmonic sums is

\begin{align}
\sum_{n \leq x}{\frac{1}{n}} &amp;= \log(x) + \gamma + O\left(\frac{1}{x}\right),
\end{align}

where $\gamma \approx 0.5772$ is the Euler-Mascheroni constant. Note that the range of the sum is over all positive integers $n$ at most $x$. The $O(1/x)$ term just means the approximation gets better as we chose larger $x$.

For $x = 50$ this formula tells us

\begin{align}
\sum_{n \leq 50}{\frac{1}{n}} &amp;\approx 4.4892.
\end{align}

If you calculate the sum exactly, the error in the approximation is about $0.00997$.
As is easily verified with your preferred CAS (I used Maple here):
$$\sum_{n=1}^{50} \frac1n = \frac{13943237577224054960759}{3099044504245996706400}$$