# series problem

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• Dec 9th 2006, 05:35 PM
synnexster
series problem
does this converge or diverge, if it converges, to what sum.

(2/3) - (4/9) + (8/27) - (16/81)

so a(n) = (-1)^n * 2^(n+1)/3^(n+1)

i know its convergent by alternating series test, but how can i figure out what sum it converges to without Sn formula?
• Dec 9th 2006, 05:48 PM
ThePerfectHacker
Quote:

Originally Posted by synnexster
does this converge or diverge, if it converges, to what sum.

(2/3) - (4/9) + (8/27) - (16/81)

so a(n) = (-1)^n * 2^(n+1)/3^(n+1)

i know its convergent by alternating series test, but how can i figure out what sum it converges to without Sn formula?

It is a geometric series,
The ratio is,
-2/3
Which is convergent since,
|-2/3|<1
Now just use the infinite geometric sum formula to find the sum.
• Dec 9th 2006, 07:05 PM
synnexster
so the sum would be a/(1-r) right?
so (2/3)/(1-(2/3)) = 2
• Dec 9th 2006, 07:23 PM
ThePerfectHacker
Quote:

Originally Posted by synnexster
so the sum would be a/(1-r) right?
so (2/3)/(1-(2/3)) = 2

Indeed, I shall wager my soul for that!