Convergence of complex series

julianabrito12

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Jul 30, 2015
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2
Hi,

I want to see of this series converge or not and understand why


Sum[sin(ik)/cos(ik)]*[(z-5-3i)^k], for k from 0 to infinity

I wanted to use the cauchy-hadamard theorem and calculate the convergence radius

for it I would say a[k]=sin(ik)/cos(ik) and do lim{abs{a[k]/a[k+1]}} as k->infinity

but I can't understand how to to a sin of a complex number and much less how to take the limits

I hope someone can help

Thank you very much

Juliana
 
Hi,

I want to see of this series converge or not and understand why


Sum[sin(ik)/cos(ik)]*[(z-5-3i)^k], for k from 0 to infinity

I wanted to use the cauchy-hadamard theorem and calculate the convergence radius

for it I would say a[k]=sin(ik)/cos(ik) and do lim{abs{a[k]/a[k+1]}} as k->infinity

but I can't understand how to to a sin of a complex number and much less how to take the limits

I hope someone can help

Thank you very much

Juliana

Hint:

e(iΘ) = cos(Θ) + i*sin(Θ)
 
As an additional hint, the sine or cosine of an imaginary number is related to the hyperbolic sine or cosine. Like so:

sin(i * k) = i * sinh(k)
cos(i * k) = cosh(k)
 
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