- #1
grog
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I've always had trouble with sequences and series, and I'm getting ready for upcoming finals now.
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.
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.