HarmonicNumber(n)

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Simon Johan

Posted: Fri Apr 25, 2008 3:34 pm

Guest

Hi,

I'm trying to understand an equation which I have seen in two books. I
states the following inequality: n/HarmonicNumber(n) <= n/log n <= n/log2 n

where the n'th harmonic number is the sum from i = 1 to n of 1/i and log is
the natural logarithm and log2 is the logarithm with base 2.

n/HarmonicNumber(n) <= n/log(n) since the n'th harmonic number is greater
than log(n) for every n, but the last part I don't get. log2(n) can be
written as log(n)/log(2) so the last inequality can be written as n/log(n)
<= log(2) * (n/log(n)) but since log(2) < 1 the inequality cannot hold.

What am I doing wrong?

Thanks in advance

Robert Israel

Posted: Fri Apr 25, 2008 3:34 pm

Guest

On Apr 25, 2:50 pm, "Simon Johan" <si...@johan.invalid> wrote:

Quote:

"Robert Israel" <isr...@math.MyUniversitysInitials.ca> wrote in message

news:rbisrael.20080425203654$4571@news.ks.uiuc.edu...

"Simon Johan" <si...@johan.invalid> writes:

I'm trying to understand an equation which I have seen in two books. I
states the following inequality: n/HarmonicNumber(n) <= n/log n <= n/log2
n

where the n'th harmonic number is the sum from i = 1 to n of 1/i and log
is

the natural logarithm and log2 is the logarithm with base 2.

n/HarmonicNumber(n) <= n/log(n) since the n'th harmonic number is greater
than log(n) for every n, but the last part I don't get.

Good, because it's false. Try n = 5.

Thanks, that's reassuring, but there must be something wrong. Maybe you
could have a look at the bottom of page 4 in this articlehttp://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19890017081_1989...

It states: sum from i = 1 to N of 1/i >= log2(N) which should be equal to
the inequality above here. Am I messing up the notation?

Either the author is ignoring constant factors, or he made a mistake.

Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Robert Israel

Posted: Fri Apr 25, 2008 3:40 pm

Guest

"Simon Johan" <simon@johan.invalid> writes:

Quote:

I'm trying to understand an equation which I have seen in two books. I
states the following inequality: n/HarmonicNumber(n) <= n/log n <= n/log2 n

where the n'th harmonic number is the sum from i = 1 to n of 1/i and log is

the natural logarithm and log2 is the logarithm with base 2.

n/HarmonicNumber(n) <= n/log(n) since the n'th harmonic number is greater
than log(n) for every n, but the last part I don't get.

Good, because it's false. Try n = 5.
--
Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Simon Johan

Posted: Fri Apr 25, 2008 4:50 pm

Guest

"Robert Israel" <israel@math.MyUniversitysInitials.ca> wrote in message
news:rbisrael.20080425203654$4571@news.ks.uiuc.edu...

Quote:

"Simon Johan" <simon@johan.invalid> writes:

I'm trying to understand an equation which I have seen in two books. I
states the following inequality: n/HarmonicNumber(n) <= n/log n <= n/log2
n

where the n'th harmonic number is the sum from i = 1 to n of 1/i and log
is

the natural logarithm and log2 is the logarithm with base 2.

n/HarmonicNumber(n) <= n/log(n) since the n'th harmonic number is greater
than log(n) for every n, but the last part I don't get.

Good, because it's false. Try n = 5.

Thanks, that's reassuring, but there must be something wrong. Maybe you
could have a look at the bottom of page 4 in this article
http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19890017081_1989017081.pdf

It states: sum from i = 1 to N of 1/i >= log2(N) which should be equal to
the inequality above here. Am I messing up the notation?

Simon Johan

Posted: Sat Apr 26, 2008 6:05 am

Guest

"Robert Israel" <israel@math.ubc.ca> wrote in message
news:d7ee9789-ef29-4f04-89c3-7bc7f34e0ef8@l25g2000prd.googlegroups.com...

Quote:

On Apr 25, 2:50 pm, "Simon Johan" <si...@johan.invalid> wrote:
"Robert Israel" <isr...@math.MyUniversitysInitials.ca> wrote in message

news:rbisrael.20080425203654$4571@news.ks.uiuc.edu...

"Simon Johan" <si...@johan.invalid> writes:

I'm trying to understand an equation which I have seen in two books. I
states the following inequality: n/HarmonicNumber(n) <= n/log n <=
n/log2
n

where the n'th harmonic number is the sum from i = 1 to n of 1/i and
log
is

the natural logarithm and log2 is the logarithm with base 2.

n/HarmonicNumber(n) <= n/log(n) since the n'th harmonic number is
greater
than log(n) for every n, but the last part I don't get.

Good, because it's false. Try n = 5.

Thanks, that's reassuring, but there must be something wrong. Maybe you
could have a look at the bottom of page 4 in this
articlehttp://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19890017081_1989...

It states: sum from i = 1 to N of 1/i >= log2(N) which should be equal to
the inequality above here. Am I messing up the notation?

Either the author is ignoring constant factors, or he made a mistake.

Thanks for your time.

Phil Carmody

Posted: Sat Apr 26, 2008 6:24 am

Guest

Robert Israel <israel@math.ubc.ca> writes:

Quote:

On Apr 25, 2:50 pm, "Simon Johan" <si...@johan.invalid> wrote:
Thanks, that's reassuring, but there must be something wrong. Maybe you
could have a look at the bottom of page 4 in this articlehttp://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19890017081_1989...

It states: sum from i = 1 to N of 1/i >= log2(N) which should be equal to
the inequality above here. Am I messing up the notation?

Either the author is ignoring constant factors, or he made a mistake.

The guy's French. The report was for NASA. It's obviously
*sabotage*!

Phil
--
Dear aunt, let's set so double the killer delete select all.
-- Microsoft voice recognition live demonstration

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