## Examples 4.2.8:

Use the Cauchy Condensation criteria to answer the following
questions:

- In the sum
,
list the terms
*a*,_{4}*a*, and_{k}*a*. Then show that this series (called the harmonic series) diverges._{2 k} - For which
*p*does the series converge or diverge ?

The sequence *{ ^{1}/_{n} }* corresponds to
the harmonic series. Therefore:

*a*_{4}= 1/4*a*_{k}= 1/k*a*_{2 k}= 1 / 2^{k}

and the last series diverges by the Divergence test. Hence, the original series also diverges.

Next, we investigate the series
for various *p*:

- If
*p < 0*then the sequence converges to infinity. Hence, the series diverges by the Divergence Test. - If
*p > 0*then consider the series

The right hand series is now a Geometric Series, so that:*=*- if
*0 < p 1*then*2*, hence the right-hand series diverges^{1-p}1 - if
*1 < p*then*2*, hence the right-hand series converges^{1-p}< 1

- if