# Convergent series and their corresponding sequences (analysis course)

• v.rad
In summary, a convergent series is an infinite series whose sum approaches a finite value as the number of terms increases. To determine if a series is convergent or divergent, various tests such as the ratio test, root test, or comparison test can be used. A convergent series has a finite sum, while a divergent series has an infinite sum. A series cannot be both convergent and divergent. The relationship between a convergent series and its corresponding sequence is that the sum of the series is equal to the limit of the sequence, and the series converges if and only if the sequence converges to the same limit.
v.rad
If a_n >= 0 for all n, and the series a_n converges, then n(a_n - a_n-1) --> 0 as n --> infinity.

Prove or disprove the statement using a counterexample.

I know that the statement is false...I am just having terrible difficultly finding a counterexample...

actually... http://www.mathhelpforum.com/math-help/f57/limit-question-indeterminate-forms-159597.html

## 1. What is a convergent series?

A convergent series is a type of infinite series where the sum of all the terms in the series approaches a finite value as the number of terms increases. In other words, as you add more and more terms, the sum of those terms gets closer and closer to a specific number.

## 2. How can I determine if a series is convergent or divergent?

To determine if a series is convergent or divergent, you can use various tests such as the ratio test, the root test, or the comparison test. These tests involve evaluating the limit of certain ratios or comparing the series to a known convergent or divergent series.

## 3. What is the difference between a convergent series and a divergent series?

A convergent series has a finite sum, meaning that as you add more and more terms, the sum of those terms approaches a specific number. On the other hand, a divergent series has an infinite sum, meaning that as you add more and more terms, the sum of those terms does not approach a specific number but rather keeps getting larger and larger.

## 4. Can a series be both convergent and divergent?

No, a series cannot be both convergent and divergent. A series can only have one of these two types of behavior. If a series is convergent, it cannot have an infinite sum, and if a series is divergent, it cannot have a finite sum.

## 5. What is the relationship between a convergent series and its corresponding sequence?

A convergent series and its corresponding sequence are closely related. The sum of a convergent series is equal to the limit of its corresponding sequence. In other words, the series converges if and only if the sequence converges to the same limit.

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