The series
= 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... is called harmonic series. It
diverges to infinity.
Proof:
We need to estimate the n-th term in the sequence of partial
sums. The n-th partial sum for this series is:
- S N = 1 + 1/2 + 1/3 + 1/4 + ... + 1/n
Now consider the following subsequence extracted from the sequence
of partial sums:
- S 1 = 1
- S 2 = 1 + 1/2
- S 4 = 1 + 1/2 + (1/3 + 1/4)
1 + 1/2 + (1/4 + 1/4) =
1 + 1/2 + 1/2 = 1 + 2/2
- S 8 = 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
In general, one can use induction (do it as an exercise) to show
that
- S 2 k
1 + k / 2 for all k
Hence, the subsequence { S 2 k }
extracted from the sequence of partial sums
{ S N } is unbounded. But then the
sequence
{ S N } can not converge either, and
must in fact diverge to infinity.
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