Infinite sum limit: (1/x)+(1/(x+1))+.......(1/2x)

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Bulkyd

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Can anyone explain to me how the following sum approaches ln2 as x approaches infinity
(1/x)+(1/(x+1))+.......(1/2x)
 
Denis said:
Seems to be something wrong...

If we use x=2, then first 2 terms are 1/2 and 1/3;
1/2 + 1/3 = .8333... so we're already > ln(2)

Can you clarify?
Click to expand...

I put it into a spreadsheet to check out the claim, and it's clearly monotonically decreasing, quite likely toward ln(2). (1.5, 1.0833, 0.95, 0.88452, ..., 0.69315?)

My question for the OP is, what is the context of the question? I can't tell whether there is some simple trick, or it could be something it takes an Euler to prove. What methods are available?

And when you ask "How", do you mean "how can I prove it?", or "how is this possible?", or "in what manner?" [monotonically!], or what?
 
Dr.Peterson said:
I put it into a spreadsheet to check out the claim, and it's clearly monotonically decreasing, quite likely toward ln(2). (1.5, 1.0833, 0.95, 0.88452, ..., 0.69315?)

My question for the OP is, what is the context of the question? I can't tell whether there is some simple trick, or it could be something it takes an Euler to prove. What methods are available?

And when you ask "How", do you mean "how can I prove it?", or "how is this possible?", or "in what manner?" [monotonically!], or what?
Click to expand...

Well, I know from some reasons that the above sum converges to ln 2 as x approaches infinity, but I don't really know how to prove it.(I know that harmonic sum of y terms approaches lny + Euler mascheroni constant, so the above must equal ln2x - lnx I.e. ln2,but I'm not allowed to use that)
 
Dr.Peterson said:
I put it into a spreadsheet to check out the claim, and it's clearly monotonically decreasing, quite likely toward ln(2). (1.5, 1.0833, 0.95, 0.88452, ..., 0.69315?)

My question for the OP is, what is the context of the question? I can't tell whether there is some simple trick, or it could be something it takes an Euler to prove. What methods are available?

And when you ask "How", do you mean "how can I prove it?", or "how is this possible?", or "in what manner?" [monotonically!], or what?
Click to expand...
Well, it should be solvable with high school level math, presuming one took calc. And yes I want a proof for it.
 
Well I know from various reasons that it approaches ln2, but I don't really know how to prove it. It was part of a question on our National Physics Olympiad and should therefore be solvable with high school presuming one took calc. So yeah, I basically want a proof for it.
 
Bulkyd said:
Well I know from various reasons that it approaches ln2, but I don't really know how to prove it. It was part of a question on our National Physics Olympiad and should therefore be solvable with high school presuming one took calc. So yeah, I basically want a proof for it.
Click to expand...
Okay. What have you tried? How far have you gotten? Where are you stuck?

We'll be glad to help, but we'll first need to see where you're getting bogged down. Please be complete. Thank you! ;)
 
Bulkyd said:
Well I know from various reasons that it approaches ln2, but I don't really know how to prove it. It was part of a question on our National Physics Olympiad and should therefore be solvable with high school presuming one took calc. So yeah, I basically want a proof for it.
Click to expand...

Being an Olympiad problem, I would not expect it to be easy, but a challenge. We'll want to see you take on that challenge, so we can help you grow from it. I myself don't see an obvious way to approach it.

On the other hand, you imply that you have omitted parts of the problem. It is possible that what you omitted might contain clues that would lead you to a solution. Can you quote the whole thing? Or maybe even give us a link to your source?

In fact, you said "I know from some reasons that the above sum converges to ln 2", rather than directly stating that the problem said this; and you said "I'm not allowed to use that", which makes me even more curious about what the problem actually says. Stating the problem in full is an important part of asking a question well.
 
Dr.Peterson said:
Being an Olympiad problem, I would not expect it to be easy, but a challenge. We'll want to see you take on that challenge, so we can help you grow from it. I myself don't see an obvious way to approach it.

On the other hand, you imply that you have omitted parts of the problem. It is possible that what you omitted might contain clues that would lead you to a solution. Can you quote the whole thing? Or maybe even give us a link to your source?

In fact, you said "I know from some reasons that the above sum converges to ln 2", rather than directly stating that the problem said this; and you said "I'm not allowed to use that", which makes me even more curious about what the problem actually says. Stating the problem in full is an important part of asking a question well.
Click to expand...
What I meant by stating that it was on the National Olympiad is that it should be doable with high school math, including single variable calculus. Anyway it was Physics Olympiad, so I doubt the question will be of any help. Still here it is,
"
Several men of equal mass are standing on a stationary railroad cart such that the combined mass of all men is equal to the mass of the empty cart. A rumour that a bomb is on the left half of the cart, however, leads to chaos and and consequently these men start jumping off the cart to the right, with equal velocities relative to the cart. Find approximately the ratio of the speed (vo) that the cart would acquire if all men jump one after the other to the speed (va) that it would acquire if all of them jump simultaneously off the cart. There is no friction between the cart and the ground. Hint: In calculating the value of the ratio, you should make use of the fact there are a large number of men."
The part where they jump off one after the other involves the summation of this series. I know that the sum converges to ln2, but have no idea how to do it with high school math, but since it was on the National Olympiad, I know it can be done that way. So yeah that pretty much it.
 
Bulkyd said:
What I meant by stating that it was on the National Olympiad is that it should be doable with high school math, including single variable calculus. Anyway it was Physics Olympiad, so I doubt the question will be of any help. Still here it is,
"
Several men of equal mass are standing on a stationary railroad cart such that the combined mass of all men is equal to the mass of the empty cart. A rumour that a bomb is on the left half of the cart, however, leads to chaos and and consequently these men start jumping off the cart to the right, with equal velocities relative to the cart. Find approximately the ratio of the speed (vo) that the cart would acquire if all men jump one after the other to the speed (va) that it would acquire if all of them jump simultaneously off the cart. There is no friction between the cart and the ground. Hint: In calculating the value of the ratio, you should make use of the fact there are a large number of men."
The part where they jump off one after the other involves the summation of this series. I know that the sum converges to ln2, but have no idea how to do it with high school math, but since it was on the National Olympiad, I know it can be done that way. So yeah that pretty much it.
Click to expand...

To my mind, "several men" is not the same as "a large number of men"; that seems odd.

But anyway, I searched for a phrase from the problem and found this explanation of it: https://crazycosmos.wordpress.com/2014/04/02/problem-5-in-indian-national-physics-olympiad-2014/ . I see no explicit mention of a limit in the first solution shown here (which uses a sum like yours); what he does is to approximate the series by an integral, which makes good sense (and, yes, the passage from sum to integral can be expressed in terms of a limit). Seeing the original problem does make a big difference. Looking for an approximation rather than a limit changes one's perspective on the question, even though it is ultimately the same thing.

I presume you see how the integral is obtained.
 
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