How can I compare two series that are same thing, but at different rates?Consider the series 1 − 1 2 + 2 3 − 1 3 + 2 4 − 1 4 + 2 5 − 1 5 I can see that if you group the terms in pairs of two, you get ( 1 − 1 2 ) + ( 2 3 − 1 3 ) + ( 2 4 − 1 4 ) + ( 2 5 − 1 5 )...= 1 2 + 1 3 + 1 4 + 1 5 ...The partial sums S 2 n = the harmonic series from n = 2 to n.

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How can I compare two series that are same thing, but at different rates?Consider the series   1  −      1    2    +      2    3    −      1    3    +      2    4    −      1    4    +      2    5    −      1    5  I can see that if you group the terms in pairs of two, you get   (  1  −      1    2    )  +  (      2    3    −      1    3    )  +  (      2    4    −      1    4    )  +  (      2    5    −      1    5    )...=       1    2    +      1    3    +      1    4    +      1    5  ...The partial sums       S          2      n      = the harmonic series from   n  =  2 to n.

elulamami · Accepted Answer

Step 1A series       ∑    n        a    n   converges, by definition, if and only if the sequence of it&#039;s partial sums   (      A    n        )    n   is convergent.Step 2Here, you have that       A          2      n        =      H    n    −  1      →          n      →      ∞            ∞. Therefore, the sequence   (      A    n        )    n   is not convergent (as otherwise every of its subsequences would be).

How can I compare two series that are same thing, but at different rates? Consider the series 1-1/2+2/3-1/3+2/4-1/4+2/5-1/5...

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