General formula for $\frac{1}{1} + \frac{1}{2}+\frac{1}{3} ... +\frac{1}{n} $

Question

We know that this series does not converge and tends to infinity but is there a general and exact formula for sum to n terms of this series

Answer 1

This is known as the $n$-th harmonic number and is denoted $H_n$.

What do you think of this formula $$H_n=\int_0^1\frac{1-x^n}{1-x}\,dx?$$

Answer 2

As already answered by Lord Shark the Unknown, this is $H_n$.

For large values of $n$, there are good approximation using their asymptotics $$H_n=\gamma +\log \left({n}\right)+\frac{1}{2 n}-\frac{1}{12 n^2}+O\left(\frac{1}{n^4}\right)$$ where $\gamma$ is Euler constant $(\approx 0.577216)$ which is very good even for small vales of $n$ as shown in the table below $$\left( \begin{array}{cccc} n & H_n & H_n \approx & \text{approximation} \\ 1 & 1 & 1.000000000 & 0.9938823316 \\ 2 & \frac{3}{2} & 1.500000000 & 1.499529512 \\ 3 & \frac{11}{6} & 1.833333333 & 1.833235361 \\ 4 & \frac{25}{12} & 2.083333333 & 2.083301693 \\ 5 & \frac{137}{60} & 2.283333333 & 2.283320244 \\ 6 & \frac{49}{20} & 2.450000000 & 2.449993653 \\ 7 & \frac{363}{140} & 2.592857143 & 2.592853705 \\ 8 & \frac{761}{280} & 2.717857143 & 2.717855123 \\ 9 & \frac{7129}{2520} & 2.828968254 & 2.828966991 \\ 10 & \frac{7381}{2520} & 2.928968254 & 2.928967425 \end{array} \right)$$