Quote:
Originally Posted by Hamlet53
Actually I think there is a mistake in Pdurrant's proof.
1+1/2+[1/3 +1/4] +[1/5+1/6+1/7+1/8] + . . . > 1 + 1/2 + (1/4 +1/4) + (1/8+1/8+1/8+1/8) + . . . = 1 +1/2 +1/2+1/2+ . .. diverges.
The point is that each group in the [] in the original series is larger than the group in the () in the second series; 1/3+1/4 > 1/4 +1/4.
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That's rather the point. We've shown that an infinite series of terms, each of which is the same or smaller than the term in our original series, diverges to infinity. Since our original series must have a larger sum than the series that diverges to infinity, it also diverges to infinity.