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Radius of convergence in series (complex plane)

1). Find the radius of convergence for each of the following power series.

Please check my solution for this problem:

a). sum ( n = 0 to infinity) a^n z^n, a is a complex number.

My solution: R( radius of convergence) = lim |a_n/a_n+1) = lim | a^n/a^(n+1)| = 1/|a|

b). Sum ( n=0 to infinity) = lim|a^(n^2)*z^n, a is complex number.

My solution: R = lim|a^(n^2)/a^(n+1)^2| = 1/|a^(2n+1)|

c). sum ( n= 0 to infinity) k^nz^n, k is an integer, k doesn't equal to 0.

My solution:
R= lim|k^n/k^(n+1)| = 1/|k|

d). sum ( n=0 to infinity) z^n!

My solution:
R = lim|1| = 1.

2).Now I am stuck with this problem...I want a detailed solution and please justify every step ( specially if you use properties of series and sums, because I am not very familiar with them)

a).Show that the radius of convergence of the power series
Sum ( n = 1 to infinity) ((-1)^n)/n (z^n*(n+1)) is 1.

b). Discuss convergence for z = 1, -1, and i.
Hint( the nth coefficient of this series is not ( (-1)^n/n).

Solution Preview

1a: your answer is correct

1b: you have an error. the correct answer is:
R = lim|a^(n^2)/a^{2(n+1)^2| = lim | 1/|a^(2n+1)| | = { either R=0, if |a| > 1; or R=infty if |a| < 1; or R=1 if |a| =1 since it would ...

Solution Summary

The radius of convergence in series for complex planes are discussed.