Question

find radius of convergence by explaining clearly

Answer 1

\[\sum_{n=0}^{\infty}x^n\]

Answer 2

x=1 must diverge because the limit as n goes to infinity is 1 which isnt 0
x>1 must diverge because the limit as n goes to infinity is infinity which isnt 0

x=-1 must diverge because the limit as n goes to infinity is -1 or 1 which means it doesnt exist and that isnt equal to 0
x<-1 must diverge because the limit as n goes to infinity doesnt exist
so u have
-1<x<1 to check now. I kno 0<x<1 converges as the geometric series
so -1<x<0 must converge because it converges absolutely as the geometric series
\[\left| (-x)^n \right|=x^n\]

Answer 3

so diameter is 1-(-1)=2
so radius is 1

Answer 4

jahtoday ,great answer thanks for ur clear explaining

Answer 5

no prob