# What is #sum_(R=1)^(N) (1/3)^(R-1)#? provide steps please.

I would separate out the exponents.

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The expression ( \sum_{R=1}^{N} \left(\frac{1}{3}\right)^{R-1} ) represents the sum of a geometric series.

- Identify the common ratio: In this case, the common ratio ( r ) is ( \frac{1}{3} ).
- Use the formula for the sum of a finite geometric series: ( S_N = \frac{a(1 - r^N)}{1 - r} ), where ( S_N ) is the sum of the first ( N ) terms, ( a ) is the first term, and ( r ) is the common ratio.

Substituting the given values:

- ( a = 1 ) (the first term)
- ( r = \frac{1}{3} ) (the common ratio)
- ( N ) (the number of terms)

[ S_N = \frac{1\left(1 - \left(\frac{1}{3}\right)^N\right)}{1 - \frac{1}{3}} ]

- Simplify the expression.

[ S_N = \frac{1 - \left(\frac{1}{3}\right)^N}{1 - \frac{1}{3}} ]

[ S_N = \frac{1 - \left(\frac{1}{3}\right)^N}{\frac{2}{3}} ]

[ S_N = \frac{3(1 - \left(\frac{1}{3}\right)^N)}{2} ]

So, ( \sum_{R=1}^{N} \left(\frac{1}{3}\right)^{R-1} = \frac{3(1 - \left(\frac{1}{3}\right)^N)}{2} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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