Summation of a series $1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}++-...$ I need to calculate the sum of the series:$T_{3n}=$ $1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}++-...$ I know that $T_{3n}=\sum_{k=1}^{n}\frac{1}{4k-3}+\frac{1}{4k-1}-\frac{1}{2k}$. And they gave a hint that $u_n=S_n-I_n$ , where $S_n=\sum_{k=1}^n \frac{1}{k}$ and $I_n=\log(n)$, converges. Can anyone give me a direction? I've tried to write $T_{3n}$ in terms of $S_n$ but without success.

Let $H_n=\sum\limits_{k=1}^n\frac{1}{k}$ 
Then as is well known $H_n-\ln n$ converges to $\gamma$
Now $$T_n=H_{4n-1}-\frac{1}{2}H_{2n-1}-\frac{1}{2}H_n$$ thus 
$$T_n-\ln (4n-1)+\frac{1}{2}\ln (2n-1)+\frac{1}{2}\ln n \to 0$$ and 
$$T_n\to \frac{3}{2}\ln 2$$
