Understanding Gaps in concept of Radius of Convergence of Power Series

by MathLover   Last Updated September 21, 2018 12:20 PM - source

I had done this topic Many time Thinking it is easy But When I today done its theorem , I come across that I had certain gaps in My Knowelege about it.
Please Help me to fill that

$\sum a_nz^n$ is power series. $1/R=limsup (a_n)^{1/n}$
1) $|Z|<R $ ,then series converges absolutely
2) $|z|>R$, then series diverges

Let $L=limsup (a_n)^{1/n}$
By defination $\forall \epsilon >0 \exists n_1\in N$ such $(a_n)^{1/n}<L+\epsilon $ Now $a_n<(L+\epsilon )^n$ i.e $a_n<(1/R+\epsilon )^n$

$|\sum a_nz^n|\leq \sum (1/R+\epsilon )^nz^n$
$(z/R+\epsilon z)^n$<1
[By 1 $|z|/R<1$ and for fix z we can vary any $\epsilon $ we wanted ]
So by Root test , RHS series is convergent
SO Original power series also converges

Similary for 2.
Is there is any gaps in my argument ?
Any Help will be appreciated

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