## 4.3. Special Series

### Definition: Harmonic Series

The series

*= 1 + 1/2 + 1/3 + 1/4 + 1/5 +*... is called harmonic series. It diverges to infinity.For an interesting application of the harmonic series, check the story about the Leaning Tower of Lire.

### Proof:

We need to estimate the*n*-th term in the sequence of partial sums.

*n*-th partial sum for this series is:

Now consider the following subsequence extracted from the sequence of partial sums:S_{N}= 1 + 1/2 + 1/3 + 1/4 + ... + 1/n

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

In general, one can use induction (do it as an exercise) to show thatS_{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

for allS_{2 k}1 + k / 2

*k*. Hence, the subsequence

*{ S*extracted from the sequence of partial sums

_{2 k}}*{ S*is unbounded. But then the sequence

_{N}}*{ S*can not converge either and must diverge to infinity.

_{N}}