Series and convergence/divergence help

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In summary, the conversation discussed problems related to convergence and divergence, specifically regarding a sequence and two series. The first problem was broken down and determined to be divergent. The second problem was suggested to use the comparison test and the third problem was suggested to use the integral test.
  • #1
Gauss177
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Hey all, I am really struggling to understand this chapter about series. These are a few problems about convergence and divergence, and I'll probably have some questions about Taylor and maclaurin series when I do those problems too.

Homework Statement


1. Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.

a_n = 9^(n+1) / 10^n

Determine whether the series is convergent or divergent:
2. 1 / n(ln n)^2, series starts at n=2 and goes to infinity

3. Find the sum of the series:
[arctan (n+1) - arctan n], series starts at n=1 and goes to infninity.


Homework Equations




The Attempt at a Solution


1. I broke the problem down to: 9(9/10)^n, and said that as n->infinity the sequence also goes to infinity, so it's divergent. Need my method checked on that one.

2. Don't know about this one

3. Examples I've seen of these kinds of problems end up being a geometric series, so I just use a/1-r to find out sum of the series. But I don't think this applies here, so what else should I do?

Thanks for any help :smile:
 
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  • #2
Gauss177 said:
1. I broke the problem down to: 9(9/10)^n, and said that as n->infinity the sequence also goes to infinity, so it's divergent. Need my method checked on that one.
Are you sure about this. For instance, the first two terms are 9*9/10=8.1, 9*81/100=7.29. The sequence seems to be decreasing. Indeed, if you have a number 'a' such that -1<a<1, then the result of raising 'a' to any power greater than one results in a number lesser than 'a'.

I would be surprised if you were asked this question and nowhere in your textbook or class notes it was mentioned that the limit of a^n is 0 for any a such that -1<a<1.
 
  • #3
1.the first series is pretty easy, it converges and its limit is 0.
evaluate it using dallamber's test, or criterion. DO you know what it states?
2. for the second one i did not solve it all, but i think that considering that the series 1/ln*n diverges, than i think that considering this you can find if your serie converges or diverges! So i think the comparison test will work on second one. As for the third i have no idea.
 
Last edited:
  • #4
For the third one, just write out the first few terms and scratch your head.
 
  • #5
And the second one needs an integral test.
 

What is a series?

A series is a mathematical expression that represents the sum of a sequence of numbers. It is written using the sigma (Σ) notation, where the index variable is shown below the sigma symbol and the expression to be summed is shown above the sigma symbol.

What is convergence and divergence in a series?

Convergence and divergence refer to the behavior of a series as the number of terms increases. A series is said to converge if the sum of its terms approaches a finite value as the number of terms increases. On the other hand, a series is said to diverge if the sum of its terms approaches infinity as the number of terms increases.

How do you determine if a series converges or diverges?

There are several tests that can be used to determine if a series converges or diverges, such as the ratio test, the root test, and the integral test. These tests involve evaluating the behavior of the series as the number of terms increases and comparing it to known patterns of convergence or divergence.

What is the difference between absolute and conditional convergence?

A series is said to have absolute convergence if the sum of the absolute values of its terms converges. On the other hand, a series is said to have conditional convergence if the sum of its terms converges, but the series of absolute values of its terms diverges. In other words, for a series to have absolute convergence, both the positive and negative terms must converge, while for conditional convergence, only the overall sum needs to converge.

Why is it important to determine if a series converges or diverges?

Determining if a series converges or diverges is important in many areas of mathematics and science, including calculus, statistics, and physics. It allows us to make accurate predictions and draw meaningful conclusions based on the behavior of the series. Additionally, knowing if a series converges or diverges can help us determine the convergence or divergence of more complex mathematical expressions that involve series.

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