I know that the harmonic series diverges $$H=\frac{1}{1}+\frac{1}{2}+...+\frac{1}{n}+...\rightarrow Diverges$$
Hence, I know that if all the terms of the harmonic series are made negative, that series also diverges $$h=-\frac{1}{1}-\frac{1}{2}-...-\frac{1}{n}+...$$
But can I write: $$h=-H$$ or is there some notation to indicate this.
I am not trying to create new maths. I am asking what is the correct notation to indicate that one series is obtained from the other by making each term negative.
As an exercise, you could prove "if any series converges with a limit L, then the same series with all elements multiplied by the same real number c converges to c*L". You can also prove "If any series converges with a limit L, and another converges with a limit L', then the series created by adding the elements converges with a limit L + L'".
That's assuming the limit is not +/- infinity. In that case a series multiplied by 0 converges to 0, while 0 times +/- infinity is not defined. And adding two series with limits +infinity and -infinity may or may not converge, and if it converges, the limit might be anything.
Your example is the first case with c = -1.