finding a formula for partial sums geometic series

**isuckatcalc** · November 8th 2009, 01:03 PM

could someone try to explain how to go about this process? i can find the sum of a series with no problem, i can find the equation for that, but i am completely lost as to how to create a formula to find the nth partial sum.
so given this sequence of numbers:
2+ 2/3 + 2/9 + 2/27 + .... how do i find the formula for the nth partial sum?

**skeeter** · November 8th 2009, 01:14 PM

Originally Posted by isuckatcalc

could someone try to explain how to go about this process? i can find the sum of a series with no problem, i can find the equation for that, but i am completely lost as to how to create a formula to find the nth partial sum.
so given this sequence of numbers:
2+ 2/3 + 2/9 + 2/27 + .... how do i find the formula for the nth partial sum?

$2\left(\frac{1}{3^0} + \frac{1}{3^1} + \frac{1}{3^2} + \frac{1}{3^3} + ... + \frac{1}{3^n}\right)<br />$

$2\sum_{i=0}^n \left(\frac{1}{3}\right)^i$

**rain** · November 8th 2009, 01:19 PM

Well, I could be wrong here, but looks to me that you're series is going $\frac{2}{3^n}$ $n=0,1,2,...$
Then the sum for the nth would be
$\sum_{i=1}^n \frac{2}{3^i} = 2\sum_{i=1}^n \frac{1}{3^i}$ (by taking the factor 2 outside of the summation)
Then the above is a geometric series
ie $2\sum_{i=1}^n \frac{1}{3^i} = 2\frac{1-\frac{1}{3}^{n+1}}{1-\frac{1}{3}}$ (I think that's correct but I can't remember the exact formula for a geometric summing series)

Edit: beaten to it

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