<s>[INST] Sums and harmonic series I found the solution of series on Wolfram Alpha 
http://www.wolframalpha.com/input/?i=sum+1%2F(2k%2B1)%2F(2k%2B2)+from+1+to+n
$ \sum\limits_{k=1}^{n} \left(\frac{1}{2k+1} - \frac{1}{2k+2}\right) = \sum\limits_{k=1}^{n} \frac{1}{(2k+1)(2k+2)} = \frac{1}{2} \left(H_{n+\frac{1}{2}} - H_{n+1} -1 + \text{ln}(4)\right)$
Can someone tell how to prove this in the form of Harmonic numbers?
 [/INST] We will make use of the following integral representation for the harmonic numbers
$$H_x = \int^1_0 \frac{1 - t^x}{1 - t} \, dt. \tag1$$
Let
$$S = \sum^n_{k = 1} \frac{1}{(2k + 1)(2k + 2)} = \sum^n_{k = 1} \left (\frac{1}{2k + 1} - \frac{1}{2k + 2} \right ).$$
Noting that
$$\int^1_0 x^{2k} \, dx = \frac{1}{2k + 1} \quad \text{and} \quad \int^1_0 x^{2k + 1} \, dx = \frac{1}{2k + 2},$$
our sum can be rewritten as
$$S = \sum^n_{k = 1} \int^1_0 (x^{2k} - x^{2k + 1}) \, dx = \int^1_0 (1 - x) \sum^n_{k = 1} x^{2k} \, dx.$$
As the finite sum appearing here is geometric, it can be summed. As
$$\sum^n_{k = 1} x^{2k} = \frac{x^2 (1 - x^{2n})}{1 - x^2},$$
one has
$$S = \int^1_0 \frac{x^2 ( 1 - x^{2n})}{1 + x} \, dx = \int^1_0 \left [\frac{x^2}{1 + x} - \frac{x^{2n + 2}}{1 + x} \right ] \, dx = I_1 - I_2.$$
The first integral is trivial. Here
$$I_1 = \int^1_0 \frac{x^2}{1 + x} \, dx = \int^1_0 \left (x - 1 + \frac{1}{1 + x} \right ) \, dx = \left [\frac{x^2}{2} - x + \ln (1 + x) \right ]^1_0 = -\frac{1}{2} + \ln (2).$$
For the second integral
\begin{align*}
I_2 &= \int^1_0 \frac{x^{2n + 2}}{1 + x} \cdot \frac{1 - x}{1 - x} \, dx = \int^1_0 \frac{(1 - x) x^{2n + 2}}{1 - x^2} \, dx.
\end{align*}
Letting $x \mapsto \sqrt{x}$ gives
\begin{align*}
S &= \frac{1}{2} \int^1_0 \frac{x^{n + 1/2} - x^{n + 1}}{1 - x} \, dx\\
&= \frac{1}{2} \int^1_0 \frac{(1 - x^{n + 1}) - (1 - x^{n + 1/2})}{1 - x} \, dx\\
&= \frac{1}{2} \int^1_0 \frac{1 - x^{n + 1}}{1 - x} - \frac{1}{2} \int^1_0 \frac{1 - x^{n + 1/2}}{1 - x} \, dx\\
&= \frac{1}{2} \left (H_{n + 1} - H_{n + 1/2} \right ),
\end{align*}
where we in the last line we have made use of (1).
Thus
$$\sum^n_{k = 1} \frac{1}{(2k + 1)(2k + 2)} = \ln (2) - \frac{1}{2} - \frac{1}{2} \left (H_{n + 1/2} - H_{n + 1} \right ),$$
as required.
</s>