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

Answer 1
I got #3//2#.

I would separate out the exponents.

#sum_(R=1)^(N) (1/3)^(R-1)#
#= sum_(R=1)^(N) (1/3)^R(1/3)^(-1)#
#= 3sum_(R=1)^(N) (1/3)^R = 3[(1/3)^1 + (1/3)^2 + . . . ]#
This is almost a geometric series, but it is missing the #R = 0# term. Therefore, we rewrite this as:
#= 3sum_(R=0)^(N) (1/3)^R - 3(1/3)^(0)#
All we did was add and subtract #3(1/3)^(0)#.
Now this is in terms of a geometric series #sum_(R=0)^(N) r^R#, and since #0 < r < 1#, this converges as
#color(blue)(3sum_(R=1)^(N) (1/3)^R)#
#= 3[1/(1 - (1//3))] - 3(1/3)^(0)#
#= 3[1/(2//3)] - 3#
#= 9/2 - 6/2#
#= color(blue)(3/2)#
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Answer 2

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

  1. Identify the common ratio: In this case, the common ratio ( r ) is ( \frac{1}{3} ).
  2. 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}} ]

  1. 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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Answer from HIX Tutor

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.

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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