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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