I does diverge, right?
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It does indeed diverge. The typical method to show this is the integral test, but it can be done in another way. Addition is associative, so you can group the terms as follows:
1 + (1/2) + [(1/3) + (1/4)] + [(1/5) + (1/6) + (1/7) + (1/8)] + ...
And compre it term-by-term to the lesser series
1 + (1/2) + [(1/4) + (1/4)] + [(1/8) + (1/8) + (1/8) + (1/8)] + ...
The second is equivalent to
1 + (1/2) + (1/2) + (1/2) + ....
which clearly diverges.
Because it is less than the original series it must "push it up" so it also diverges.
1 + (1/2) + [(1/3) + (1/4)] + [(1/5) + (1/6) + (1/7) + (1/8)] + ...
And compre it term-by-term to the lesser series
1 + (1/2) + [(1/4) + (1/4)] + [(1/8) + (1/8) + (1/8) + (1/8)] + ...
The second is equivalent to
1 + (1/2) + (1/2) + (1/2) + ....
which clearly diverges.
Because it is less than the original series it must "push it up" so it also diverges.
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yes it diverges