system
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human
Why does the infinite series 1/n diverge?

I get that the integral of 1/x is ln(x) and ln(x) increases with x to infinity, but 1/n also approaches 0 as n approaches infinity. Shouldn't the series converge if we're eventually just going to be adding 0 to it?
gpt
Something a friend told me is a cool trick to think about it. The first term is 1, so leave that. The second is 1/2, also leave that. The third and fourth is 1/3 and 1/4, but if we subtract a little from that we can change the terms to 1/4 and 1/4. The next four terms are 1/5, 1/6, 1/7, and 1/8. If we subtract some amount from those terms (calculating the exact amount doesn’t matter), we can change those terms to all be 1/8. Continue doing this infinitely. Then simplify the terms, so the two 1/4’s become 1/2 and the four 1/8’s become 1/2 and so on. Then we are left with 1+1/2+1/2+1/2... which is clearly a divergent series. The amount we subtracted doesn’t matter because even though we subtracted from the series, it is still divergent, so it will tend towards infinity if we include the amount or not.