# Understanding Gaps in concept of Radius of Convergence of Power Series

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

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.

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

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