I'm focusing on summability theory/summation methods/resummation theory and related topics, in other words methods to give meaningful sum to divergent series. First I recall that a method is called linear if:
$$\sum_{n=0}^{+\infty} a_n+b_n=\sum_{n=0}^{+\infty} a_n+\sum_{n=0}^{+\infty} b_n$$
holds.
A well-know inequality is that: $|a+b|\leq |a|+|b|$ so I guess that if all of the series converge there is no harm (correct me if I'm wrong) in assume that results:
$$\sum_{n=0}^{+\infty} |a_n+b_n|\leq \sum_{n=0}^{+\infty} |a_n|+\sum_{n=0}^{+\infty} |b_n|$$
But is this true for divergent series too (if I'm working with a linear method) ? I guess no so maybe the question should be: is there any linear summation method (standard summation doesn't obviously count) for which this property holds ? However I'm skeptical of this too 'cause a divergent sum of positive terms can be summed to a negative value so what if I consider this:
$$|\sum_{n=0}^{+\infty} |a_n+b_n||\leq |\sum_{n=0}^{+\infty} |a_n||+|\sum_{n=0}^{+\infty} |b_n||$$