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.

Hint: Add two hopefully known converging series term-wise:
$$
\begin{array}{cccccccccccc}
&1&-&\dfrac12&+&\dfrac13&-&\dfrac14&+&\dfrac15&-&\dfrac16&+&\dfrac17&-&\dfrac18&+&\dfrac19&-&\dfrac1{10}&+&\cdots\\
\\
&0&+&\dfrac12&+&0&-&\dfrac14&+&0&+&\dfrac16&+&0&-&\dfrac18&+&0&+&\dfrac1{10}&+&\cdots\\
\\
\hline
\\
=&1&+&0&+&\dfrac13&-&\dfrac12&+&\dfrac15&+&0&+&\dfrac17&-&\dfrac14&+&\dfrac19&+&0&+&\cdots
\end{array}
$$
