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Challenges for newbs

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worker201:
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?

Laukev7:
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.

Pathos:
lol infinity.

worker201:
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.

TheQuirk:
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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