Thursday, 7 January 2016

limits - Find the convergence radius of series




I need to find the convergence radius of the following series:
n=0cosin×zn


Since
cosin=coshn=ein+ein2

therefore, using D'Alembert's property I find the limit as
liman+1an=ei(n+1)+ei(n+1)2×zn×z×2(ein+ein)zn=limz×ei(n+1)+ei(n+1)ein+ein=z

Therefore the radius should be 1. But the answer on the book is 1e.
What's wrong with my solution?


Answer



You have two errors here. The first is that

coshn=en+en2


not what you wrote (which is cosn. To remember: cosh is unbounded on R, so you have "true" exponentials, while cos is bounded, so you have eix which is bounded). So you need to compute
limnen+1+e(n+1)en+en
and not limnei(n+1)+ei(n+1)ein+ein (where you also made a mistake, the second: this limit does not exist, anyway!).



So let's compute (1):
en+1+e(n+1)en+en=en(e+e(2n+1))en(1+e2n)==e+e(2n+1)1+e2nne+01+0=e.


Therefore,
limnan+1an=ze

which explains why the radius is 1e.


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