Complex Variables Convergence of Summations
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A) Prove that sum(z^n/n) converges at every point of the unit circle except z=1 although this power series has R=1.
b) Use partial fractions to determine the following closed expression for
c_n
c_n=((1+sqrt5/2)^n+1 - (1-sqrt5/2)^n+1)/sqrt5
Ps. Here c_n are Fibonacci numbers defined by c_0=1, c_1=1,....
c_n=c_n-1 + c_n-2, for n=2,3.....
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Solution Summary
The solution to the following problem is given in detailed in the attached solution file.
Prove that sum(z^n/n) converges at every point of the unit circle except z=1 although this power series has R=1.
Use partial fractions to determine the following closed expression for c_n
c_n=((1+sqrt5/2)^n+1 - (1-sqrt5/2)^n+1)/sqrt5
Solution Preview
Please see the attached solution file which contains detailed step by step solution to the stated problems.
Solution:
The radius of convergence is 1.
Also for z = 1 , and the series ...
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