Find the values of x for which the series converges, then find the sum of the series for those values of x. `sum_(n=0)^oo(x-1)^n/5^n`

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

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This is a geometric series `sum_(n=0)^oo r^n` , where `r=(x-1)/5.` As with any geometric series, this series converges if and only if `|(x-1)/5|<1.` Solving for `x` gives `-4<x<6.`

The series converges if and only if `-4<x<6.`

For these values of `x,` the first term is always `1,` and the ratio between terms is `r.` The sum is thus