# Another necessary condition

• jk22
In summary, the Cauchy definition of convergence of a series involves the condition that the partial sums of the series approach a limit as the number of terms increases. This can be rewritten in terms of a positive decreasing sequence, and taking the limit can provide another necessary criterion for convergence. However, it is difficult to generalize this criterion to series with different types of terms, as seen with the example of the alternating harmonic series. While it may be possible to extend this criterion to positive term series, the proof remains elusive.

#### jk22

Starting from the Cauchy definition of convergence of a series :

$$\forall N,\epsilon>0,\exists N_0 | k>N_0\Rightarrow |\underbrace{\sum_{n=1}^{N+k}u_n-\sum_{n=1}^k u_n}_A |<\epsilon$$

rewriting A in terms and considering a positive decreasing sequence :

$$A\Rightarrow \epsilon>u_{N+k}+\ldots +u_{k+1}>Nu_{N+k}$$

one finds by taking the limit another necessary criterion :

$$\lim_{n\to\infty}n u_n=0$$.

This implies for example that the harmonic series cannot converge.

My question is why we don't see this at school but only the condition $$u_n\to 0$$ ?

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You are imposing the additional restriction that you have a positive decreasing sequence. The "Divergence Test" that is taught in school applies to all series.

Do you think it's possible to generalize this to any kind of series ?

jk22 said:
Do you think it's possible to generalize this to any kind of series ?
Take the alternating harmonic series.
It converges.
But, it wouldn't, according to the simplest form of generalization of your criterion.
The upshot is that it probably is VERY difficult to generalize this Cauchy criterion into an independent necessary criterion for convergence for series of a more general type than those with positive, decreasing terms.

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Right.
I thought maybe it is possible to generalize to positive term series (not forcedly decreasing).

The sum of the terms then becomes : $$Nu_m$$ where u_m is the minimum of the terms between N+k and k, but then remains to prove that m tends towards N when N tends to infinity, but I have no idea how to do that.

## 1. What is meant by "Another necessary condition"?

"Another necessary condition" refers to an additional factor or requirement that is essential for a particular outcome or result to occur. In other words, it is something that must be present in addition to other conditions in order for a desired outcome to be achieved.

## 2. How is "Another necessary condition" different from other types of conditions?

"Another necessary condition" differs from other types of conditions, such as sufficient conditions, in that it alone is not enough to ensure a specific outcome. Instead, it must be combined with other conditions in order to produce the desired result.

## 3. Can you give an example of "Another necessary condition" in a scientific context?

One example of "Another necessary condition" in a scientific context could be the presence of oxygen in addition to water and sunlight for photosynthesis to occur in plants. Without oxygen, the other necessary conditions would not be sufficient for photosynthesis to take place.

## 4. How do scientists determine if something is "Another necessary condition"?

Scientists determine if something is "Another necessary condition" by conducting experiments and observing the effects of adding or removing the potential condition on the outcome. If the outcome is not achieved without the additional condition, it can be considered another necessary condition.

## 5. Why is it important to identify "Another necessary condition" in scientific research?

Identifying "Another necessary condition" is important in scientific research because it allows for a more complete understanding of the factors that contribute to a particular phenomenon or outcome. It also helps to refine and improve scientific theories and models by accounting for all necessary conditions. Additionally, recognizing "Another necessary condition" can also have practical applications, such as in the development of new technologies or treatments.