I'm a little confused on geometric series.(adsbygoogle = window.adsbygoogle || []).push({});

My book says that a geometric series is a series of the type: n=1 to ∞, ∑ar^{n-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 = 1/2^{n}

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?

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# I Geometric Series Confusion

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