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September 1st, 2012, 10:15 PM   #1
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Divergence and Asymptotes


1. Prove or disprove : diverges.

2. Prove or disprove : has an asymptote.

Thank you for any help,

Balarka

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September 2nd, 2012, 01:17 AM   #2
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Re: Divergence and Asymptotes

HI mathbalarka:

I assume is the polylogarithm?.

The polylog is defined as

But the problem simply states . Perhaps I am misunderstanding, but which particular polylog is to be used?.

In this case, it would be x=1 and mean . Then, we would have the limit

as part of the problem?.

Another notation would be

i.e. the dilogarithm:

which, when x=1, would be the classic .

In other words, we have to know which polylog we are dealing with. My apologies if I am not getting something obvious.
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September 2nd, 2012, 01:27 AM   #3
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Re: Divergence and Asymptotes

No, is the offset logarithmic integral
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September 2nd, 2012, 01:30 AM   #4
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Re: Divergence and Asymptotes

For 1, proving it disproving it diverges would probably use a technique similar to proving the Euler-Mascheroni constant converges. I don't know how to do that. It would be somehow showing that diverges. The Euler-Mascheroni constant has x in place of log(x). The differences probably do not decrease fast enough as it's logarithmic.
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