# Series - Testing for Convergence / Divergence

• steelphantom
In summary, the conversation discusses several series and the methods used to determine their convergence or divergence. The first series is assumed to diverge and the comparison test is used to find a smaller divergent series. The second series is larger than a convergent series and therefore cannot be proven to converge. The third series is analyzed using Stirling's formula to show that it does not approach 1 as n becomes large, suggesting that it converges.
steelphantom
I have a few series which I'm having trouble proving whether they converge or diverge. I know the following tests for convergence: comparison test, ratio test, n-th term test, and root test. Here are the series and what I have tried so far:

$$\sum$$ n -1 / n2 : I'm assuming this series diverges, since it behaves like 1/n, which also diverges. I'm trying to use the comparison test to see if I can find a "smaller" series which also diverges, but coming up blank. I tried the ratio test to no avail, since it gives 1.

$$\sum_{n=2}^\infty$$ 1 / (n + (-1)n)2 : I'm really not sure where to begin with this one. The (-1)n is really throwing me off. I'm assuming this converges.

And finally,

$$\sum$$ n! / nn : I tried the ratio test, canceling out the factorial and getting the ratio of nn / (n + 1)n. This limit seems to be 1, so the ratio test doesn't really help me here. Any suggestions?

Thanks for any help!

Last edited:
For the first one, how about saying (n-1)/n^2>=(n/2)/n^2? Notice that doesn't work for n=1, but that's ok. You can always ignore any finite number of terms. For the second one, your series is LARGER than 1/(2n)^2. That's no good. Being larger than a convergent series doesn't tell you much. How about 1/(n/2)^2? The last one is more delicate. Do you know Stirling's formula?

For the third one, n^n/(n+1)^n does NOT approach 1 as n becomes large.

Ok, thanks for the help on the first two! I noticed that my series was larger than 1/(2n)^2 right after I posted. Dumb mistake. I don't know Stirling's formula. Is there another way to go about this?

jhicks, if n^n/(n+1)^n doesn't approach 1, I can only assume it approaches a number less than 1, in which case the series would converge. How could I show this limit is less than 1?

jhicks is right. (n/(n+1))^n=1/((n+1)/n)^n=1/(1+1/n)^n. Does that look familiar?

Ah, I see it now. Thanks a lot!

## 1. What is the purpose of testing for convergence/divergence in a series?

The purpose of testing for convergence/divergence in a series is to determine whether the series will approach a finite limit (converge) or approach infinity (diverge) as more terms are added. This is important in understanding the behavior and properties of the series, as well as its potential applications.

## 2. How is the convergence/divergence of a series determined?

The convergence/divergence of a series is determined by evaluating its limit as the number of terms approaches infinity. If the limit exists and is finite, the series is said to converge. If the limit does not exist or is infinite, the series is said to diverge.

## 3. What are some common tests for convergence/divergence in a series?

Some common tests for convergence/divergence in a series include the ratio test, the comparison test, and the integral test. These tests use various mathematical techniques to analyze the behavior of the series and determine its convergence or divergence.

## 4. What is the difference between absolute and conditional convergence in a series?

Absolute convergence refers to a series where the sum of the absolute values of its terms converges, while conditional convergence refers to a series where the sum of its terms converges, but not necessarily the sum of their absolute values. Absolute convergence is a stronger form of convergence and ensures that the series will converge regardless of the order in which its terms are added.

## 5. How can testing for convergence/divergence be applied in real-world scenarios?

The concept of convergence/divergence is widely used in various fields of science and engineering, such as physics, economics, and computer science. It can help predict the behavior of systems or processes over time and determine their stability or instability. For example, in physics, testing for convergence/divergence is used to analyze the behavior of infinite series in mathematical models of physical phenomena.

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