Can someone explain to me why one serie is convergent and the other one divergent? both seem to lead to 0 to me? Thx a lot for your help
http://www.math.uni-siegen.de/numeri...ine/img883.gif
http://www.math.uni-siegen.de/numeri...ine/img888.gif
Can someone explain to me why one serie is convergent and the other one divergent? both seem to lead to 0 to me? Thx a lot for your help
http://www.math.uni-siegen.de/numeri...ine/img883.gif
http://www.math.uni-siegen.de/numeri...ine/img888.gif
The general term goes to zero as n becomes large, but in neither case
can the sum be zero since they are the sums of positive terms and therefore
the sum must be bigger than the first term.
The first is the harmonic series, which can be shown to diverge by grouping
the terms:
$\displaystyle
\sum_1^{\infty} \frac{1}{k} = 1 + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4} \right)+\left( \frac{1}{5} + \frac{1}{6}+\frac{1}{7} + \frac{1}{8} \right)+ ...
$
Each group after the first has $\displaystyle 2^{n-1}$ terms and the smallest term is $\displaystyle 1/2^n$ and so each is greater $\displaystyle 1/2$ so:
$\displaystyle
\sum_1^{\infty} \frac{1}{k} > 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2}+ ...
$
which shows that the series is divergent.
Showing that the second converges is fairly simple, the integral test should do
the job. Determing what it sums to is more tricky but Robin Chapman give a
number of methods here.
RonL