I am trying to find a way to prove that $$\dfrac 11 + \dfrac 12 + \dfrac 13 + \dfrac 14 + \cdots \color{red}{-} \dfrac 18 + \cdots$$
where the pattern repeats every $8$ terms. Knowing about the Riemann Series Theorem, I am a little hesitant about manipulating conditionally convergent series at all. Granted that the harmonic series diverges, is the following a valid way to prove my series diverges?
$$\dfrac 11 + \dfrac 12 + \dfrac 13 + \cdots + \dfrac 17 – \dfrac 18 + \cdots = \sum_{n=1}^{\infty} \dfrac 1n – 2 \cdot \dfrac 18\sum_{n=1}^{\infty} \dfrac 1n $$
$$=\dfrac 34 \sum_{n=1}^{\infty} \dfrac 1n$$
Since the harmonic series diverges, so does $\dfrac 34$ times it.
Best Answer
Your idea is a good one, but, as you suspected, you need to be more careful about this sort of manipulation of conditionally convergent series.
One way to carry out your argument correctly, but with only minor changes, is by looking at partial sums:
Let's write $$a_n=\begin{cases}1,&\text{ if }n\text{ is not a multiple of 8} \\-1,&\text{ if }n\text{ is a multiple of 8},\end{cases},$$ so that your series is $\sum_{n=1}^\infty \frac{a_n}{n}.$
Then for any natural number $N,$
\begin{align} \sum_{n=1}^{8N}\frac{a_n}{n} &= \sum_{n=1}^{8N}\frac1{n}-2\sum_{n=1}^N \frac1{8n} \\&=\sum_{n=1}^{8N}\frac1{n}-\frac1{4}\sum_{n=1}^N \frac1{n} \\&\ge\sum_{n=1}^{8N}\frac1{n}-\frac1{4}\sum_{n=1}^{8N} \frac1{n} \\&\quad\scriptsize{\quad\text{(because we can only be subtracting *more* positive numbers)}} \\&=\frac3{4}\sum_{n=1}^{8N}\frac1{n}, \end{align}
which approaches $\infty$ as $N$ approaches $\infty,$ since the harmonic series diverges.