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#