Both series are divergent, so can we conclude that they are divergent by addition and subtraction? I don't remember. What is the principle?

The sum of two divergent series may converge or diverge. such as

1)∑( 1/n) and ∑( 1/n? -1/n) diverges, but the sum converges;

2)σ( 1/n) and σ( 1/n? +1/n) are divergent.

If a series is convergent, the terms of this series will definitely tend to zero. Therefore, any series whose term does not tend to zero is divergent. However, convergence is a stronger requirement than this: not every series whose term tends to zero converges. One of the counterexamples is harmonic series.

Extended data:

The function that a convergent series maps to its sum is linear, so according to Hahn-Barnach theorem, it can be deduced that this function can be expanded into the sum of a series that can be bounded with any part. This fact is generally not very useful, because many of these expansions are incompatible with each other, and because the existence of this operator proves that it appeals to axiom of choice or its equivalent forms, such as Zuo En Lemma, which are unstructured.

Divergent series, as a field of analysis, is essentially concerned with clear and natural techniques, such as Abel and cesaro and borell and other related objects. The appearance of Wiener-Taubel theorem indicates that this branch has entered a new stage, which leads to an unexpected connection between Banach algebra and summability in Fourier analysis.

As a numerical technique, the summation of divergent series is also related to interpolation and sequence transformation. Examples of this technique include Padre approximation, Levin-like sequence transformation and sequence mapping related to renormalization technique of higher-order perturbation theory in quantum mechanics.

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