Infinite Series Comparison Test

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  • #1
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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?
 
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  • #2
mathman
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You are comparing (1) and (2), asserting terms of (1) are less than terms of (2). This is not true, even for the next term. For (1) the next term is 1/14 while for (2) it is 1/16. As you continue the terms of (2) get a lot smaller than the terms of (1).

I hope you were not being facetious.
 
  • #3
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I hope you were not being facetious.
Heheh, apparently I wasn't. I guess I didn't see all sides of the argument.
Still...is grouping like that valid?...it doesn't work on a series such as:
1 + (-1) + 1 + (-1) + ... or is that only because this one alternates in sign?
 
  • #4
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Heheh, apparently I wasn't. I guess I didn't see all sides of the argument.
Still...is grouping like that valid?...it doesn't work on a series such as:
1 + (-1) + 1 + (-1) + ... or is that only because this one alternates in sign?
Rearrangement of terms in a series should only be attempted if

(1) all terms are nonnegative, or

(2) the series is absolutely convergent.

As discussed in a different thread, the Riemann Series Theorem indicates rearrangement does not guarantee equality in other cases.

--Elucidus

EDIT: (1) can be expanded to include "eventually nonnegative."
 
  • #5
1,179
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Rearrangement of terms in a series should only be attempted if

(1) all terms are nonnegative, or

(2) the series is absolutely convergent.

As discussed in a different thread, the Riemann Series Theorem indicates rearrangement does not guarantee equality in other cases.

--Elucidus

EDIT: (1) can be expanded to include "eventually nonnegative."
Interesting...Thanks for the replies.
 

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