How do you find the radius of convergence $Sigma 5x^n$ from $n=[0,oo)$ ?

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Answer 1 · 2017-09-06T10:00:03

The radius of convergence is $R=1$

We apply the ratio test

Let $a_n=5x^n$

Then

$lim_(x->oo)|(5x^(n+1))/(5x^n)|=lim_(x->+oo)|x|=1*|x|$

The series converge fo $|x|<1$

But, we must check for convergence when $|x|=1$

When $x=-1$ , $=>$ , $5sum_(n=0)^oo(-1)^n$ diverges by the

geometric test criteria as $|r|>=1$

When $x=1$ , $=>$ , $5sum_(n=0)^oo(1)^n=5$ diverges as every infinite sum of a non-zero constant

The interval of convergence is $-1 < x < 1$

How do you find the radius of convergence Sigma 5x^nSigma 5x^n from n=[0,oo)n=[0,oo)?