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# Find all z such that the series 1 / (1+ |z|2)n converges

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 z such that the series (n / (n+1)) (z / (2z+1))n converges

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 z such that the series (z / (2z+1))n converges

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 z such that the series (-1)n / (2n+1) • ((1-z) / (1+z))n converges

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 z such that the series (2z+3)n / (n log (n+1)) converges

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 z such that the series (-1)n(z-1)n / n converges

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 z such that the series (z-1)n / (n+2)! converges

Find all complex numbers such that the series

converges.

Consider,

Hence, the series converges for all complex .

# Find all z such that the series (1 + 1/(5n+1))n2 |z|17n converges

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 z such that the series zn / n1/2 * log ((2n+1) / n) converges

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 z such that the series (-1)n / (z+n) converges

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.