Quote:
Originally Posted by Hamlet53
That series, the harmonic series, diverges. Can't put the symbols to make this simple but the area under the curve of the function 1/x from x=1 to x=infinite, or the integral from x=1 to x = infinite of 1/x dx is less than the sum of the rectangles of height 1/n and width 1 from n=1 to n=infinite. The mentioned integral diverges and therefore so does the series.
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You're right that it diverges, but your proof isn't very elegant.
Here's the elegant proof.
Divide the series into groups as follows:
1 + (1/2 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9 + 1/10 + 1/11 + 1/12 + 1/14 + 1/14 + 1/15 + 1/16) + ....
Replace each term in each group with the smallest term in the group. Note that this reduces the sum of the series.
1 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + (1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16) + .....
Note that each and every group sums to 1/2
1 + (1/2) + (1/2) + (1/2) + .....
Which clearly diverges to infinity. Our original sequence has a larger sum than this, and so also diverges to infinity.
Q.E.D.
Hamlet53 gets to set the next question.