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I'm a little confused on geometric series.
My book says that a geometric series is a series of the type: n=1 to ∞, ∑arn-1
If r<1 the series converges to a/(1-r), otherwise the series diverges.
So let's say we have a series: n=1 to ∞, ∑An, with An = 1/2n
An can be re-written as (1/2)n, which apparently makes it a geometric series with r=1/2. This converges to 1/(1-1/2) = 1/(1/2) = 2.
However, I was under the assumption that I was supposed to factor out a 1/2 to make the series ∑1/2(1/2)n-1.
This would converge to (1/2)/(1-1/2) = (1/2)/(1/2) = 1.
Why are these different? Which one is correct?
My book says that a geometric series is a series of the type: n=1 to ∞, ∑arn-1
If r<1 the series converges to a/(1-r), otherwise the series diverges.
So let's say we have a series: n=1 to ∞, ∑An, with An = 1/2n
An can be re-written as (1/2)n, which apparently makes it a geometric series with r=1/2. This converges to 1/(1-1/2) = 1/(1/2) = 2.
However, I was under the assumption that I was supposed to factor out a 1/2 to make the series ∑1/2(1/2)n-1.
This would converge to (1/2)/(1-1/2) = (1/2)/(1/2) = 1.
Why are these different? Which one is correct?