Does the series converge or diverge? (r=1..inf)

In summary, the conversation discusses the convergence or divergence of the series \sum (1-\frac{1}{r})^{r^2}, where r is a variable that ranges from 1 to infinity. The conversation mentions trying to use the nth-term test for divergence, but it is determined that the limit of the sequence does not necessarily equal zero for the series to converge. The conversation concludes with a discussion about the contrapositive of the nth-term test and how it is logically equivalent to the original statement.
  • #1
sid9221
111
0
[tex] \sum (1-\frac{1}{r})^{r^2} [/tex]

Does this converge or diverge.(r=1..inf)

I have tried the following but do not think it is adequate(or correct for that matter)

[tex] (1-\frac{1}{r})^r (1-\frac{1}{r})^r = (1-\frac{1}{r})^{r^2} [/tex]

and [tex] lim (1-\frac{1}{r})^r -> \frac{1}{e} [/tex]

thats given from a previous part of this question.

So
[tex] lim (1-\frac{1}{r})^r (1-\frac{1}{r})^r -> \frac{1}{e^2} [/tex]

Hence converges ?(As the limit exists)
 
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  • #2
According to the nth-term test for divergence, the limit=1, hence the series diverges.

P.S. Is that problem part of a question? If earlier sections of the problem are related to this question, you should post the whole thing.
 
  • #3
Well its part of a group of questions but seemingly unrelated.

I've checked on wolfram that this series converges, so I don't think the test you quoted maybe valid...?

Wolfram say's to do a ratio test but that's not feasible by hand(at least to me)
 
  • #4
First, you have an error in your algebra. ##x^n x^n = x^{2n}## not ##x^{n^2}##.

Second, you're confusing the convergence of the series ##\sum a_n## with the convergence of the sequence ##a_n##. Just because the sequence converges doesn't mean the series converges.

Finally, what must ##a_n## converge to if the series is to converge?
 
  • #5
Don't know what I was thinking !

So for it to converge the limit has to go to zero ? (Don't think the contrapostive of the non-null test is true though ?)

Say I put [tex] \sum[({1-\frac{1}{r}})^r]^2 [/tex]

Than took the limit of the inner part would I get [tex] \frac{1}{e^2} [/tex] It still would not be equal to zero...?
 
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  • #6
vela said:
Finally, what must ##a_n## converge to if the series is to converge?

I'm not sure that i follow this. Is it based on any particular test? How do you arrive at that conclusion? I'm sorry for asking, but it seems that i messed up my understanding of sequences v/s series. I can relate this to AST, where the limit of [itex]a_n[/itex] needs to be zero for the series to converge.
 
  • #7
sid9221 said:
So for it to converge the limit has to go to zero ? (Don't think the contrapostive of the non-null test is true though ?)
The contrapositive has to be true. It's logically equivalent to the original statement. The nth-term test says if ##\displaystyle \lim_{n\to\infty}a_n\ne 0## then the infinite series ##\displaystyle \sum_n a_n## will not converge. The contrapositive would be: if the series converges, then the limit of an has to be 0.

What I think you're thinking is if an goes to 0, it doesn't necessarily mean that the series converges. That is correct.

Say I put [tex] \sum[({1-\frac{1}{r}})^r]^2 [/tex]

Than took the limit of the inner part would I get [tex] \frac{1}{e^2} [/tex] It still would not be equal to zero...?
Right, so...

sharks said:
I'm not sure that i follow this. Is it based on any particular test? How do you arrive at that conclusion? I'm sorry for asking, but it seems that i messed up my understanding of sequences v/s series. I can relate this to AST, where the limit of [itex]a_n[/itex] needs to be zero for the series to converge.
You've forgotten about the nth-term test. It's probably the very first test mentioned when you began to study series.
 
  • #8
vela said:
You've forgotten about the nth-term test. It's probably the very first test mentioned when you began to study series.

I know about the nth-term test but its definition says that as long as the limit of the sequence does not equal zero, the series has to diverge. This theorem says nothing about convergence. If the limit is equal to zero, then according to the nth-term test, the series could either converge or diverge.

vela said:
Finally, what must ##a_n## converge to if the series is to converge?

I'm not sure this is applicable to the nth-term test for divergence.
 
  • #9
Read what I wrote above about the contrapositive.
 
  • #10
vela said:
Read what I wrote above about the contrapositive.

OK, you simply started with the end-result (that the series converges) and then backtracked to deduce that the limit has to be equal to zero.
 
  • #11
vela said:
The contrapositive has to be true. It's logically equivalent to the original statement. The nth-term test says if ##\displaystyle \lim_{n\to\infty}a_n\ne 0## then the infinite series ##\displaystyle \sum_n a_n## will not converge. The contrapositive would be: if the series converges, then the limit of an has to be 0.

I'm getting confused as well, an obvious example would be:

[tex] a_n = \frac{1}{n} -> 0 [/tex] as n->infinity

But

[tex] \sum \frac{1}{n} [/tex]

diverges ?

So how is the contrapositive true.

If your saying that we already know that the series is converging than that statement is true, that maybe different but in this case we don't know if it converges or diverges.
 

1. What is the definition of convergence of a series?

The convergence of a series refers to the behavior of the terms in a sequence as the number of terms increases. It describes whether the sum of the terms approaches a finite value (converges) or diverges (grows without bound).

2. What are the different types of convergence for series?

The three main types of convergence for series are absolute convergence, conditional convergence, and divergence. Absolute convergence occurs when the series converges regardless of the order of the terms. Conditional convergence occurs when the series converges, but its rearrangement can lead to a different sum. Divergence occurs when the series does not converge to a finite value.

3. How is the convergence of a series determined?

The convergence of a series can be determined by applying various convergence tests, such as the ratio test, the root test, and the comparison test. These tests use the properties of the terms in the series to determine whether it will converge or diverge.

4. What is the importance of convergence in mathematics?

The concept of convergence is crucial in mathematics as it allows us to study infinite sequences and series. Convergent series have well-defined sums, and this allows us to make calculations and draw conclusions about the behavior of the sequence or series. Convergence also plays a key role in many real-world applications, such as in physics and engineering.

5. What are some common misconceptions about convergence of series?

One common misconception is that if the terms in a series approach zero, then the series must converge. This is not always true, as there are cases where the series can still diverge. Another misconception is that if a series converges, then its terms must approach zero. However, there are convergent series where the terms do not approach zero, such as the harmonic series. It is important to use convergence tests to determine the behavior of a series rather than relying on these misconceptions.

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