Understanding the Convergence of a Sequence: Exploring the Limit of 1/2

In summary, the conversation discusses the convergence of a sequence defined by fractions and the limit of the sequence. The question is raised about how the index appears in every term and whether it is a sequence or series. It is explained that the limit of the sequence is zero and the limit of the sequence of partial sums is 1/2. The conversation also touches on the possibility of thinking of a sequence as a series, but it is considered an awkward approach.
  • #1
quasar987
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Apparently (according to my textbook), the sequence defined by

[tex]\left\{\frac{1}{n^2}+\frac{2}{n^2}+...+\frac{n-1}{n^2}\right\}[/tex]

converges towards 1/2, i.e. has 1/2 as a limit.

How could that be?! It seems to me that as n approaches infinity, all the fractions fall to zero. What is it I'm missing?
 
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  • #2
The question is if they tend to zero faster than their number grow towards infinity.
 
  • #3
:eek:

Is there a way to find this analytically?
 
  • #4
quasar987 said:
:eek:

Is there a way to find this analytically?
Sure; you may write the partial sum as:
[tex]\frac{1}{n^{2}}(1+2+++n-1)=\frac{1}{n^{2}}\frac{n(n-1)}{2}[/tex]
 
  • #5
arildno said:
Sure; you may write the partial sum as:
[tex]\frac{1}{n^{2}}(1+2+++n-1)=\frac{1}{n^{2}}\frac{n(n-1)}{2}[/tex]

Gauss's sum again! Damn! You guys are smart, are you all doctors in mathematics or physics?
 
  • #6
Some of them are. :) Don't worry about it, I feel the same way you do all the time.
 
  • #8
quasar987 said:
Apparently (according to my textbook), the sequence defined by

[tex]\left\{\frac{1}{n^2}+\frac{2}{n^2}+...+\frac{n-1}{n^2}\right\}[/tex]

converges towards 1/2, i.e. has 1/2 as a limit.

Hold on a second. How is it that the index appears in every term when you list out the series?

Also, the above is a series, not a sequence.

How could that be?! It seems to me that as n approaches infinity, all the fractions fall to zero. What is it I'm missing?

The limit of the sequence is zero.
The limit of the sequence of partial sums is 1/2.
 
  • #9
Tom Mattson said:
Hold on a second. How is it that the index appears in every term when you list out the series?

Also, the above is a series, not a sequence.

Tom, it's precisely the fact that the index "n" appears in each of the terms that makes this a sequence, and not a series, as it's given.

[tex]a_n=\sum_{i=1}^{n-1}\frac{i}{n^2}[/tex]

It's the limit of [tex]a_n[/tex] he's after. Since each of the terms in the sum is dependant on n, you can't break it into a series as I suspect you are thinking of doing.


You can of course think of any sequence as a series, by setting [tex]b_1=a_1, b_n=a_n-a_{n-1}[/tex], then [tex]a_n=\sum_{i=1}^{n}b_i[/tex], but that can be an awkward thing to do. In this case we'd find [tex]b_n=\frac{1}{2n(n+1)}[/tex], but I don't think that's what you were getting at?
 
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  • #10
I really do know better than that...

Do me a favor and just ignore me for the rest of the night...
 
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What is the definition of convergence of a sequence?

Convergence of a sequence refers to the concept in mathematics where a sequence of numbers approaches a specific value or limit as the number of terms in the sequence increases.

How is convergence of a sequence determined?

Convergence of a sequence is determined by calculating the limit of the sequence, which is the value that the terms in the sequence get closer and closer to as the number of terms increases.

What is the difference between a convergent and a divergent sequence?

A convergent sequence approaches a specific limit as the number of terms increases, while a divergent sequence does not have a specific limit and either approaches infinity or oscillates between values.

What is the importance of convergence of a sequence?

Convergence of a sequence is important in mathematics as it allows for the analysis of infinite processes and helps determine the behavior of a sequence as the number of terms increases. It also has applications in real-world scenarios, such as in physics and engineering.

How can convergence of a sequence be tested?

There are several tests that can be used to determine the convergence of a sequence, such as the comparison test, ratio test, and root test. These tests involve evaluating the terms in the sequence and comparing them to known convergent or divergent sequences.

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