Find all complex numbers such that the series
converges.
This is a geometric series
Therefore it converge if and only if
Hence, the series converges for all except
.
Find all complex numbers such that the series
converges.
This is a geometric series
Therefore it converge if and only if
Hence, the series converges for all except
.
Find all complex numbers such that the series
converges.
First, we have
We know from the previous exercise (Section 10.20, Exercise #44) that the series
converges if and only if
Therefore, the series converges if since both terms in the sum converge. The series diverges if
since
diverges and
converges. Finally, the series diverges if
since
.
Find all complex numbers such that the series
converges.
This is a geometric series. It converges if
and diverges if
If then the series also diverges since
in this case.
Find all complex numbers such that the series
converges.
Let
Then the series converges if converges. But, since
is a geometric series, it converges if
If we write this inequality holds if and only if
.
On the other hand,
and so
hence, the series diverges.
Therefore, the series converges for all with
.
Find all complex numbers such that the series
converges.
First, we consider
So, by the ratio test, the series converges if and diverges if
. If
with
then the series also converges by Dirichlet’s test. Hence, the series converges for
with
.
Find all complex numbers such that the series
converges.
If then we apply the Leibniz rule, where
is monotonically decreasing and has limit 0 as
. If
with
, then we apply Dirichlet’s test with
to conclude that the series converges. If
then
and the series diverges since it equals the harmonic series.
Find all complex numbers such that the series
converges.
Consider,
Hence, the series converges for all complex .
Find all complex numbers such that the series
converges.
We look to apply the root test, letting
Then we have
Since this series converges if this limit is less than 1 and diverges if it is greater than 1, we have the series is absolutely convergent for
Find all complex numbers such that the series
converges.
First, we simplify the expression as follows,
Then, we consider the first series,
Let and use the ratio test,
This series converges if and diverges if
. If
and
then the series also converges by Dirichlet’s test. Finally, if
, then the series is
which diverges.
Next, we know
Thus,
Hence, this series converges anywhere converges. Therefore, the sum
converges for with
.
Find all complex numbers such that the series
converges.
First, the series is not absolutely convergent for any since
By limit comparison with this always diverges.
Then, if for any positive integer
the series is convergent since
is monotonically decreasing and
. If
for some positive integer
, then the series is not defined since it is undefined for that term.