I've always had trouble with sequences and series, and I'm getting ready for upcoming finals now.(adsbygoogle = window.adsbygoogle || []).push({});

There's an example in my calculus text that says:

Show that the harmonic series [tex]\sum[/tex]1/n is divergent.

The solution states: For this particular series it's convenient to consider the partial sums s2,s4,s8,s16,s32,...and show that they become large.

s1=1

s2=1 + (1/2)

s4= 1 + 1/2 + (1/3 + 1/4) > 1+ 1/2 + (1/4 + 1/4) = 1+ 2/2

s8=1 + 1/2 + (1/3 + 1/4) +(1/5 + 1/6 + 1/7 + 1/8) > 1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) = 1 +1/2 +1/2 +1/2 = 1+3/2

so the pattern becomes s2n > 1+ n/2

which shows that s2n -> infinity as n -> to infinity and so {sn} is divergent. Therefore the harmonic series diverges.

What I don't understand is why the terms get substituted, for smaller ones (1/3+1/4 becomes 1/4+1/4, etc). If I understood why they were doing that I would understand the rest of it, but there's no explanation in the book, unless it was covered in an earlier section.

Any help understanding would be most appreciated.

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# Question about series convergence/divergence

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