Looks like the Harmonic series

In summary: In each term, we are adding a larger number than the previous term. This means that the sum is still increasing, but at a decreasing rate.
  • #1
Punkyc7
420
0
does let z[itex]_{n}[/itex]=

[itex]\frac{1}{n+1}[/itex]+[itex]\frac{1}{n+2}[/itex]+[itex]\frac{1}{n+3}[/itex]+...+[itex]\frac{1}{2n}[/itex]

does z[itex]_{n}[/itex] converge or diverge..


I want to say it diverges because it looks like the Harmonic series
 
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  • #2


yep I am positive
 
  • #3


Punkyc7 said:
does let z[itex]_{n}[/itex]=

[itex]\frac{1}{n+1}[/itex]+[itex]\frac{1}{n+2}[/itex]+[itex]\frac{1}{n+3}[/itex]+...+[itex]\frac{1}{2n}[/itex]

does z[itex]_{n}[/itex] converge or diverge..


I want to say it diverges because it looks like the Harmonic series
Each term in the sum above is >= 1/(2n), which is the smallest term, and there are n terms. This means that zn >= n/(2n) = 1/2.

Also, each term in the sum is <= 1/(n + 1), which is the largest term, and there are still n terms. This means that zn <= n/(n + 1).

One of the tests you have learned is applicable here.
 
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  • #4


does that mean it converges?
 
  • #5


Write down a few terms in your sequence.
z1 = 1/2
z2 = 1/3 + 1/4
z3 = 1/4 + 1/5 + 1/6

What can you say about
[tex]\lim_{n \to \infty} z_n~?[/tex]

Note that I added some to my previous response.
 
  • #6


[tex]
\begin{array}{rcl}
z_{n + 1} - z_{n} & = & \left( \frac{1}{n + 2} + \ldots + \frac{1}{2 n} + \frac{1}{2 n + 1} + \frac{1}{2 n + 2} \right) \\

& - & \left(\frac{1}{n + 1} + \frac{1}{n + 2} + \ldots + \frac{1}{2 n} \right) \\

& = & \frac{1}{2 n + 1} + \frac{1}{2 n + 2} - \frac{1}{n + 1} \\

& = & \frac{1}{2 n + 1}- \frac{1}{2 n + 2} > 0
\end{array}
[/tex]

Post #4 holds the other clue.
 
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  • #7


as n goes to infinity isn't the limit 1? because of zn <= n/(n + 1).
 
  • #8


About all you can say is that 1/2 <= zn <= 1, but that doesn't mean that the limit has to be either of the endpoints.
 
  • #9


since its bounded and increasing it converges doesnt
 
  • #10


Punkyc7 said:
as n goes to infinity isn't the limit 1? because of zn <= n/(n + 1).

no. look at, for example:
[tex]
z_{2 n} = \left( \frac{1}{2n + 1} + \ldots + \frac{1}{3 n} \right) + \left(\frac{1}{3 n + 1} + \ldots + \frac{1}{4 n}\right) \le \frac{n}{2 n + 1} + \frac{n}{3 n + 1} = \frac{n (5 n + 2)}{(2 n + 1)(3 n + 1)}
[/tex]
[tex]
z_{2 n} \le \frac{5 n^2 + 2 n}{6 n^2 + 5 n + 1} = \frac{1}{\frac{6 n^2 + 5 n + 1}{5 n^2 + 2 n}} = \frac{1}{\frac{6}{5} + \frac{\frac{13 n}{5} + 1}{n (5 n + 2)}} \le \frac{1}{\frac{6}{5}} = \frac{5}{6}
[/tex]

Thus, there is a subsequence that is definitely not within a neighborhood [itex]\epsilon = 1/5[/itex] of 1. So, 1 is not a limit of the sequence.

Punkyc7 said:
since its bounded and increasing it converges doesnt
yes, but it's actually decreasing.
 
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  • #11


ok since its monotonic decreasing and bounded it is convergent
 
  • #12


Dickfore said:
yes, but it's actually decreasing.

It is NOT decreasing, it's increasing. I'd suggest to Punkyc7 not to believe everything everyone tells you.
 
  • #13


Punkyc7 said:
ok since its monotonic decreasing and bounded it is convergent

yes, and the limit is somewhere between 1/2 and 5/6. Actually, the exact sum is expressible via elementary functions. Do you know what it is?
 
  • #14


Dick said:
It is NOT decreasing, it's increasing. I'd suggest to Punkyc7 not to believe everything everyone tells you.

read post #6.
 
  • #15


Dickfore said:
read post #6.

I did. And I have no idea what you are doing. z_1=1/2, z_2=7/12, z_3=37/60. Now you read post #6 and tell me what's wrong with it.
 
  • #16


there's an extra term -1/n that does not belong there.
 
  • #17


Dickfore said:
there's an extra term -1/n that does not belong there.

Perhaps. I didn't read it THAT carefully. But I do know z_n is an increasingly accurate estimate to log(2). And all of the z_n are underestimates. Think of it as like an integral estimate.
 
  • #18


Dick said:
Perhaps. I didn't read it THAT carefully.
I corrected it.

Dick said:
But I do know z_n is an increasingly accurate estimate to log(2). And all of the z_n are underestimates. hink
of it as like an integral test.

Look at my post #13.
 
  • #19


Dickfore said:
I corrected it.



Look at my post #13.

Ok, so you know what it is. That's good. I'd still suggest to Punkyc7 to think more about the problem. And maybe for you to maybe provide less details until Punkyc7 does that.
 
  • #20


The original sum is a sum of positive terms. If we are always adding a positive number, the sum must be increasing.
 
  • #21


HallsofIvy said:
The original sum is a sum of positive terms. If we are always adding a positive number, the sum must be increasing.

But, we are not adding a positive number in addition to the same positive numbers from the previous term. Thus, your logic is flawed.
 

1. What is the Harmonic series?

The Harmonic series is a mathematical concept that involves adding an infinite number of terms, each of which is the reciprocal of a positive integer. It is represented by the formula 1 + 1/2 + 1/3 + 1/4 + 1/5 + ...

2. What does it mean when something "looks like the Harmonic series"?

When something "looks like the Harmonic series," it means that it follows a similar pattern to the Harmonic series, where each term is the reciprocal of a positive integer. This can be seen in various natural and man-made phenomena, such as the series of overtones in music or the frequencies of electromagnetic waves.

3. What are some examples of things that look like the Harmonic series?

Some examples of things that look like the Harmonic series include the ratios of the lengths of the strings in a musical instrument, the frequency of beats in a human heartbeat, and the spacing of energy levels in an atom.

4. Why is the Harmonic series important in science?

The Harmonic series is important in science because it is a fundamental concept that helps us understand the relationships between different quantities in the natural world. It also has important applications in fields such as physics, mathematics, and music theory.

5. How is the Harmonic series related to other mathematical concepts?

The Harmonic series is related to other mathematical concepts such as geometric series, Fibonacci sequence, and the Riemann zeta function. It also has connections to the study of prime numbers and the distribution of energy levels in quantum systems.

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