I am trying to solve $\sum\limits_{k=1}^n\frac{2k+1}{k(k+1)}$ using summation by parts:
$\sum u\Delta v=uv-\sum Ev\Delta u$
$u = 2x+1, \Delta v=1/x(x+1)=(x-1)_{-2}$
$v=-(x-1)_{-1}=-1/x$
$\Delta u = 2, Ev = -1/(x+1)$
$\sum\frac{(2x+1)}{x(x+1)}\delta x=\frac{-2x-1}{x}+2\left(\sum x_{-1}\delta x\right)$
Is my solution correct up to this point? If it is, what is the anti-difference $\sum x_{-1}\delta x$? This seems to be the finite equivalent to $\int x^{-1}dx=ln(x)$.
On
I'm not sure this fully answers your questions, but it is easier to solve this problem without summation by part, simply by partial fraction expansion: $$ S_n = \sum_{k=1}^{n}\frac{1}{k} + \sum_{k=1}^{n}\frac{1}{k+1}=2H_n-1+\frac{1}{n+1}=2H_n-\frac{n}{n+1} $$ where $H_n$ is n'th harmonic number
Your computations are correct, which is easy to see by differencing.
The anti-difference of $x_{-1} = \frac{1}{x+1}$ is known a Harmonic number: $$ \sum_x \frac{1}{x+1} = H_x $$ where $H_n = \sum_{k=1}^n \frac{1}{k}$.