# Series and convergence/divergence

• Niles
In summary, the conversation discusses determining whether the series \sum\limits_{n = 2}^{\inf } {\frac{{( - 1)^n (n^2 + 1)^{1/2} }}{{n\ln (n)}}} converges and suggests using the ratio or root test, as well as the Leibniz test for alternating series. There is also a question about the first condition of the Leibniz test, specifically if it refers to the limit of a_n for n -> infinity.
Niles

## Homework Statement

Determine whether the series converges:

$$\sum\limits_{n = 2}^{\inf } {\frac{{( - 1)^n (n^2 + 1)^{1/2} }}{{n\ln (n)}}}$$

## The Attempt at a Solution

Which test must I use? I thought of using the integral test, but it seems a little too hard. Are there other possibilities?

Try either the ratio or root test. Neither would be very easy to calculate, but...

Well, the $(-1)^n$ should give you an easy test...

BTW: \infty is $\infty$

Should I just take:

$$\sum\limits_{n = 2}^{\inf } {\frac{{( - 1)^n (n^2 + 1)^{1/2} }}{{n\ln (n)}}}^{(1/n)}$$ and take the limit of this?

Its an alternating series, what about the Leibinitz test?

I looked at the Leinitz test - on wikipedia they write about the first condition:

"If the sequence a_n converges to 0, and .." - does this mean the limit of a_n for n -> infinity?

Last edited:

## 1. What is a series?

A series is a sum of an infinite sequence of numbers, typically denoted by the sigma notation ∑. Each term in the sequence is added together to determine the value of the series.

## 2. What is the difference between convergence and divergence of a series?

A convergent series is one whose terms approach a finite value as the number of terms increases. In other words, the sum of the series is a finite number. On the other hand, a divergent series is one whose terms do not approach a finite value and the sum of the series is either infinity or negative infinity.

## 3. How do you determine if a series is convergent or divergent?

To determine the convergence or divergence of a series, we can use various tests such as the ratio test, the root test, or the integral test. These tests compare the terms of the series to well-known sequences that either converge or diverge, ultimately determining the convergence or divergence of the series.

## 4. What is the importance of series and convergence/divergence in mathematics?

Series and convergence/divergence have various applications in mathematics, especially in calculus and analysis. They are used to approximate functions, evaluate integrals, and solve differential equations. Additionally, they are important in understanding the behavior of infinite sequences and the concept of infinity.

## 5. 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 properties. However, it is possible for a series to be conditionally convergent, meaning that it is convergent but not absolutely convergent. In other words, the series converges but the absolute values of the terms do not converge.

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