# How do you find the sum of the finite geometric sequence of #Sigma 2(-1/4)^n# from n=0 to 40?

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The sum of the finite geometric sequence Σ 2(-1/4)^n from n=0 to 40 can be found using the formula for the sum of a finite geometric series: (S = a \frac{{1 - r^{n+1}}}{{1 - r}}), where (a) is the first term, (r) is the common ratio, and (n) is the number of terms. In this case, (a = 2), (r = -1/4), and (n = 40). Substituting these values into the formula yields:

(S = 2 \frac{{1 - (-1/4)^{40+1}}}{{1 - (-1/4)}})

(S = 2 \frac{{1 - (-1/4)^{41}}}{{1 + 1/4}})

(S = 2 \frac{{1 - (-1/4)^{41}}}{{5/4}})

(S = \frac{{8}}{{5}}(1 - (-1/4)^{41}))

(S ≈ \frac{{8}}{{5}})

So, the sum of the finite geometric sequence Σ 2(-1/4)^n from n=0 to 40 is approximately (\frac{{8}}{{5}}).

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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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