integral test Watch

Chewwy
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#1
Report Thread starter 13 years ago
#1
http://plus.maths.org.uk/issue19/fea...ndex-gifd.html

i don't understand how part of equation (15) comes about. how come 1 + ln(n) < (the sum)? isn't the second comparison, which isn't written down:

sum < ∫(1/x)dx
sum < ln(x)

where does the one come from???

using ln(x), we get ln(x+1) < sum < ln(x) which can't even be possible...
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bob_54321
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#2
Report 13 years ago
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it's because lnx has the y axis as an asymptote so rather than saying that the sum is less than the integral for all of it, you know that the first number in the sum is 1 and so you use that instead. that doesnt come accross very clearly in my explanation.

the one is basically just adding on the first bit of the sum, the rectangle of area 1 because otherwise the integral is messy. i think thats what youre asking.
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Chewwy
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#3
Report Thread starter 13 years ago
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ah ha. thankyou.
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bob_54321
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#4
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you appear to have understood, but i just realised, its nothing to do with y=lnx being an asymptote, its y=1/x. and is it me or is the label under the graph under equation (15) wrong, it is 1/n underneath 1/x isn't it . . . not 1/1+x. minor detail.
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