I need to find the convergence radius of the following series:
∞∑n=0cosin×zn
Since
cosin=coshn=ein+e−in2
therefore, using D'Alembert's property I find the limit as
liman+1an=ei(n+1)+e−i(n+1)2×zn×z×2(ein+e−in)zn=limz×ei(n+1)+e−i(n+1)ein+e−in=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+e−n2
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
limn→∞en+1+e−(n+1)en+e−n
So let's compute (1):
en+1+e−(n+1)en+e−n=en(e+e−(2n+1))en(1+e−2n)==e+e−(2n+1)1+e−2n→n→∞e+01+0=e.
Therefore,
limn→∞an+1an=z⋅e
which explains why the radius is 1e.
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