Sum of 1 + 1/2 + 1/3 +.... + 1/n

There is no simple closed form. But a rough estimate is given by

$$ \sum_{r=1}^n \frac{1}{r} \approx \int_{1}^n \frac{dx}{x} = \log n $$

So as a ball park estimate, you know that the sum is roughly $\log n$. For more precise estimate you can refer to Euler's Constant.

There is no simple expression for it.

But it is encountered so often that it is usually abbreviated to $H_n$ and known as the $n$-th Harmonic number.

There are various approximations and other relations which you can find in Wikipedia under Harmonic Number or in the question Jose Santos referenced in the comments.

For example, $$H_n=G_n-(n+1)\lfloor\frac{G_n}{n+1}\rfloor$$ where $$G_n=\frac{{n+(n+1)!\choose n}-1}{(n+1)!}$$

But that kind of thing is more of a curiosity than a useful expression!

One can write $$1+\frac12+\frac13+\cdots+\frac1n=\gamma+\psi(n+1)$$ where $\gamma$ is Euler's constant and $\psi$ is the digamma function.

Of course, one reason for creating the digamma function is to make formulae like this true.