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I read a proof for showing that the harmonic series is a diverging one. This particular one used a comparison test:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + ... + 1/16 + ...

1/2 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + 1/16 +... + 1/16 + ...

Each term in the second series is < or = to the corresponding one in the first...and the proof is that grouping terms in the second series gives 1/2 + 1/2 + 1/2 + 1/2 + ..., which diverges. The problem I'm having here is accepting the validity of grouping the terms to produce a sum of continuous 1/2's...there was also the case, in which:

(1) 1/2 + 1/5 + 1/8 + 1/11 + ... is less than

(2) 1 + 1/2 + 1/4 + 1/8 + ... and because (2) converges as a geometric series with r < 1, by the comparison test, series (1) converges. However:

(3) 1/3 + 1/6 + 1/9 + 1/12 + ...can be written as 1/3(1 + 1/2 + 1/3 + 1/4 + ...), which diverges (as a harmonic series). (3) is less than (1) term by term and should converge by the comparison test, but it doesn't according to observation (I'm trying to find the error in the argument just stated since it was stated that there is one). If grouping were valid, then wouldn't there be a noticeable flexibility in the validity of the comparison test?

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + ... + 1/16 + ...

1/2 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + 1/16 +... + 1/16 + ...

Each term in the second series is < or = to the corresponding one in the first...and the proof is that grouping terms in the second series gives 1/2 + 1/2 + 1/2 + 1/2 + ..., which diverges. The problem I'm having here is accepting the validity of grouping the terms to produce a sum of continuous 1/2's...there was also the case, in which:

(1) 1/2 + 1/5 + 1/8 + 1/11 + ... is less than

(2) 1 + 1/2 + 1/4 + 1/8 + ... and because (2) converges as a geometric series with r < 1, by the comparison test, series (1) converges. However:

(3) 1/3 + 1/6 + 1/9 + 1/12 + ...can be written as 1/3(1 + 1/2 + 1/3 + 1/4 + ...), which diverges (as a harmonic series). (3) is less than (1) term by term and should converge by the comparison test, but it doesn't according to observation (I'm trying to find the error in the argument just stated since it was stated that there is one). If grouping were valid, then wouldn't there be a noticeable flexibility in the validity of the comparison test?

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