anyone still able to help here please?
For this example, I have many questions about this problem, can someone please answer my questions, and then add anything else about this problem i may need to know.
heres the worked problem: http://img143.imageshack.us/img143/66/untitledfh4.png
questions:
Is this an infinite series?
Why do we select the first part of the Series as Sn? What is Sn?
..I understand that 1/2n is the starting number and the limit for this series,
and that it is greater than any Sn, however, why do we add + 1/2?
(for example, this part: 1/2n >= Sn + 1/2)
..also, then why does the above equation then change to: 1/2n >= n * 1/2n = 1/2
Is this series divergent because the statement: 0 >= 1/2 is false.
Is that the only reason it is divergent?
I would really appreciate it, if someone could answer these questions the best they can.
Thanks for any help here.
I'm trying to undersand your question, but I'm confused by the notation. It *looks* wrong. This is probably why no one has responded yet.
For example, is Sn supposed to be read as "S sub n"? In other words, was it written as a capital S with a subscript n?
Also, in your work you show a summation:
SUM{i=1 -> 2n} [1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... + 1/(2n)]
Clearly this summation is wrong. The sum is from i = 1 to i = 2n, but you've written the summation as a sum of term of n instead of being in terms of i. I think it's safe to assume you meant to write the summation:
SUM{i=1 -> 2n} [1/i]
No. It's a sum from 1 to 2n.Is this an infinite series?
If I've accurately guessed the notation you should be using, S_n, is the short way of saying "the series sum from 1 to n" whereas S_(2n) is "the series sum from 1 to 2n"Why do we select the first part of the Series as Sn? What is Sn?
You misread this. It did not say 1/(2n) > Sn + 1/2, it said(for example, this part: 1/2n > Sn + 1/2)
S_(2n) = SUM{i=1 -> 2n} [1/i] = 1 + 1/2 + 1/3 + ... + 1/(2n) >= S_n + 1/2
Shortened, this says
S_(2n) > S_n + 1/2
Also, the "..." indicated that the terms go on.
In this case,
1 + 1/2 + 1/3 + ... + 1/(2n) is the same as saying
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + ... + 1/(n - 2) + 1/(n - 1) + 1/n + 1/(n + 1) + 1/(n + 2) + ... + 1/(2n - 2) + 1/(2n - 1) + 1/(2n)
Even here, I had to use the "..." because it would be impossible to write out every term of this series if n is not defined.
1/(2n) is the starting number and the limit for the series? I'm not sure what you mean by this. However, as far as them adding the + 1/2, here's what they were saying:..I understand that 1/2n is the starting number and the limit for this series,
and that it is greater than any Sn, however, why do we add + 1/2?
S_(2n) > S_n + 1/2
In other words, the sum of the series from 1 to 2n adds up to a number larger than the sum of the series from 1 to n plus + 1/2. (Even by adding 1/2 to the sum of the series from 1 to n, the sum from 1 to 2n will still be greater.)
Be very careful with your notation (I think I've made this point quite a few times). In this case, you need to be wary of "order of operations."..also, then why does the above equation then change to: 1/2n >= n * 1/2n = 1/2
n * 1/2n = 1/2n^2
However, the problem you have is not "n * 1/2n," it's "n * 1/(2n)"
n * 1/(2n) = n/(2n) = 1/2 ... the n's reduced.
Besides this, your question is basically, how do we get 1/2? I'll show you.
S_n = 1 + 1/2 + 1/3 + ... + 1/(n - 1) + 1/n
S_(2n) = 1 + 1/2 + 1/3 + ... + 1/(n - 1) + 1/n + 1/(n + 1) + 1/(n + 2) + ... + 1/(2n - 1) + 1/(2n)
Notice that S_n and S_(2n) both have the terms:
1 + 1/2 + 1/3 + ... + 1/(n - 1) + 1/n
We can subtract S_n from S_(2n) and all of the terms I just wrote will disappear, leaving us with:
S_(2n) - S_n = 1/(n + 1) + 1/(n + 2) + ... + 1/(2n - 1) + 1/(2n)
There are two observations I can make about this summation.
1) I know there are "n" terms there. How do I know this? Because S_(2n) has "2n" terms total, S_n has "n" terms total. If we subtract S_(2n) from S_n, we are left with "2n - n" = "n" terms.
2) I know that 1/(2n) is smaller than 1/(2n - 1) and 1/(2n) is smaller than 1/(2n - 2) and 1/(2n) is smaller than 1/(n + 2) and ... etc. 1/(2n) is a smaller number than all of the rest of the terms in that summation above.
Given these two facts, I know that:
S_(2n) - S_n > n*(1/(2n)) AND n*(1/(2n)) = 1/2
Therefore,
S_(2n) - S_n > 1/2
If I add S_n to both sides, I get
S_(2n) > S_n + 1/2 ... look familiar?
Simply put, this series is not divergent because it is not infinite.Is this series divergent because the statement: 0 >= 1/2 is false.
Is that the only reason it is divergent?
yes, you are correct its a subscript. I'll use S_n from now on. my appoligy.
also, this work was provided for me in lecture, by a professor. this is unfortunate, because the work is wrong!?
i'll try to follow and understand your work here.
thanks for all the help so far.
awesome. good explanation.If I've accurately guessed the notation you should be using, S_n, is the short way of saying "the series sum from 1 to n" whereas S_(2n) is "the series sum from 1 to 2n"
I've seen here and earlier above where you've used ... + 1/(n - 1) + 1/nBesides this, your question is basically, how do we get 1/2? I'll show you.
S_n = 1 + 1/2 + 1/3 + ... + 1/(n - 1) + 1/n
S_(2n) = 1 + 1/2 + 1/3 + ... + 1/(n - 1) + 1/n + 1/(n + 1) + 1/(n + 2) + ... + 1/(2n - 1) + 1/(2n)
and ... + 1/(2n -1) + 1/(2n)
..however, i did not see any of this in the example, why do you start by adding terms in the series, and then change to start subtracting? ex: (2n - 1)
Theres alot going on in this problem, could you tell me in a sentence or two, whats the goal for this problem, to prove what? or maybe what kind of series test it is. I'm missing the logic i think.
I appreciate all the help you've provided so far, I'm struggling with this type of math.