Stop Microsoft
Miscellaneous => Programming & Networking => Topic started by: worker201 on 21 August 2005, 03:20
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Show that the infinite series:
1/1 + 1/2 + 1/3 + 1/4 .... is convergent. Meaning, that even though it never ends, it gets closer and closer and closer to a certain value. What is that value?
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That series does not converge. This is a textbook problem which even has a name; it's called the harmonic series.
From what I remember, the principle is this: you start with 1/1, which is a whole number.
Then, you add up 1/2, which gives 3/2.
Adding up 1/3 and 1/4 gives a number which is higher than the previous number, 1/2. Adding up the four next numbers close to 1/4 gives a number which is still higher than 1/2, and so on. Each sequence of number adds up to an ever-increasing number.
Therefore, there can be no convergence.
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lol infinity.
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Well, according to the C program I wrote yesterday, which runs the harmonic series all the way to 1/8000, it does in fact seem convergent. I'm going to run it to 1/16000 tonight, to check my theory.
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1 + 1/2 + ... + 1/n approximately equals the integral of 1/x, which is ln(x), which is an increasing function. It doesn't converge, but it seems to because it increases slowly (the tangent line to ln(8000) has a slope of 0.000125).
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Well, at least we got some higher mathematics onto this forum!
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it's not convergent, it's the numerical equivalent of that thing of throwing a dart at a dartboard, the dart will never get there because it will always have to travel half the distance in the interim.
actually it's not that at all, because in that example, both time and distance get halved, whereas in this example only one variable does, follow?
anyway, the sequence does get closer and closer to zero, however it will never reach zero, because the figure on the bottom of the fraction can never be absolutely high (so as to make the resultant fraction equal to zero, yes?)
well, i know what i mean anyway...
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Yes, yes, except this sequence is increasing, not decreasing.
Lim x--> infinity = infinity
(but it sure as hell looks like it is going to run out of steam before it hits 10...)
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oh i see what you're doing
in that case, it's like the representation of a snail's shell.