Convergence/Divergence of a Series

  • Thread starter d.tran103
  • Start date
  • Tags
    Series
In summary: This series converges. In summary, if the series is convergent, then the limit is zero. If the series is divergent, then the limit does not exist or is different from 0.
  • #1
d.tran103
39
0
Determine whether or not the series is convergent or divergent. If it is convergent, find its sum.

1) 1/3 + 1/6 + 1/9 + 1/12 + 1/15 + ...


inf.
2) E (n-1)/(3n-1)
n=1

Okay, I'm having trouble with determining the convergence/divergence. My book states that, "If the series

inf.
E Asubn
n=1

is convergent, then lim as n goes to inf. asub n=0."

and it also states that, "If lim as n goes to inf. asubn does nto exist or if lim as n goes to inf. does not equal 0, then the series

inf.
E Asubn
n=1
is divergent.

----
Here's where I'm having trouble. I set up the sum series for the first question as:
inf.
E Asubn 1/(3n)
n=1

Then I took the limit as n goes to inf. of 1/(3n) and got 0. So by the book, wouldn't this make it convergent? However my book has the answer as being divergent.

Then for the second question, I took the limit as n goes to inf. of (n-1)/(3n-1) to be 1/3. So wouldn't this make it convergent as well? The book has the answer as being divergent.

If someone would articulate the convergent/divergent test a little better it would be greatly appreciated. Thanks!
 
Physics news on Phys.org
  • #2
The theorem says:

IF the series converges, THEN the limit is zero.

You argue, that the limit is zero, thus the series converges. But that's the converse implication and it does not hold.

For example:
IF my name is Alex, THEN my name begins with an A.

What you do is: my name begins with an A so my name must be Alex.
This is of course invalid.
 
  • #3
Okay thanks, that makes sense now. But if I find that the limit is 0, then it can be convergent or divergent. How do I tell from there?

inf.
E 1/3n
n=1

lim 1/3n = 0
n-->inf.
 
  • #4
Try comparing it with the series [tex]\sum_{n=1}^\infty \frac{1}{n}[/tex].
 
  • #5


I understand your confusion with the convergence/divergence test for series. Let me break it down for you in a way that may be easier to understand.

First of all, a series is convergent if the sum of its terms approaches a finite number as the number of terms increases. On the other hand, a series is divergent if the sum of its terms becomes infinitely large as the number of terms increases.

Now, let's apply this to the first series. We can rewrite it as:

1/3 + 1/6 + 1/9 + 1/12 + 1/15 + ... = 1/3(1 + 1/2 + 1/3 + 1/4 + 1/5 + ...)

As you can see, this is a harmonic series, which is known to be divergent. This means that as the number of terms increases, the sum of the terms becomes infinitely large. Therefore, the first series is divergent.

For the second series, we can rewrite it as:

E (n-1)/(3n-1) = E (1-1/n)/(3-1/n)

As n approaches infinity, the term (1-1/n) approaches 1 and (3-1/n) approaches 3. Therefore, the series becomes:

E 1/3 = inf.

Since the sum of the terms becomes infinitely large, this series is also divergent.

In summary, the key to determining convergence or divergence of a series is to look at the behavior of the terms as the number of terms increases. If the terms approach a finite number, the series is convergent. If the terms become infinitely large, the series is divergent. I hope this helps to clarify the concept for you.
 

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

Convergence and divergence refer to the behavior of a series, or a sum of infinitely many terms. A series is said to converge if the sum of its terms approaches a finite value as the number of terms increases. In contrast, a series is said to diverge if the sum of its terms becomes infinitely large as the number of terms increases.

2. How can 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 comparison test, the ratio test, and the integral test. These tests involve finding the limit of a function or comparing the series to a known convergent or divergent series.

3. What is the importance of knowing the convergence or divergence of a series?

Knowing the convergence or divergence of a series is important in many areas of mathematics and science, as it can help us understand the behavior of functions and make predictions about their values. It is also essential in determining the validity of mathematical calculations and proofs.

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 behaviors. However, there are some series that do not have a defined behavior and are said to be indeterminate.

5. What is the difference between absolute and conditional convergence?

Absolute convergence refers to the convergence of the absolute values of the terms in a series, while conditional convergence refers to the convergence of the series itself. A series can be absolutely convergent but not conditionally convergent, but a conditionally convergent series must also be absolutely convergent.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
928
  • Topology and Analysis
Replies
3
Views
907
Replies
3
Views
2K
  • Topology and Analysis
Replies
0
Views
140
Replies
2
Views
1K
  • Topology and Analysis
2
Replies
44
Views
5K
Replies
2
Views
1K
  • Topology and Analysis
Replies
16
Views
3K
Replies
4
Views
184
Replies
15
Views
2K
Back
Top