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...

racmanovcf

racmanovcf

Answered question

2022-10-18

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.

Answer & Explanation

elulamami

elulamami

Beginner2022-10-19Added 22 answers

Step 1
A series n a n converges, by definition, if and only if the sequence of it's partial sums ( A n ) n is convergent.
Step 2
Here, 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).

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