sum_(k=1)^(oo)4((1)/(3))^(k-1)

Recognize series type: We recognize that the series is a geometric series with the first term $ a = 4 $ (since when $ k = 1 $, the term is $ 4 \cdot 1 $) and the common ratio $ r = \frac{1}{3} $. The sum of an infinite geometric series can be found using the formula $ S = \frac{a}{1 - r} $, where $ |r| < 1 $ for the series to converge.Check convergence: We check the common ratio to ensure that the series converges. Since $ |r| = |\frac{1}{3}| = \frac{1}{3} < 1 $, the series converges.Apply sum formula: We apply the formula for the sum of an infinite geometric series: $ S = \frac{a}{1 - r} $. Substituting $ a = 4 $ and $ r = \frac{1}{3} $, we get $ S = \frac{4}{1 - \frac{1}{3}} $.Simplify denominator: We simplify the denominator: $ 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} $.Calculate sum: We calculate the sum: $ S = \frac{4}{\frac{2}{3}} $. To divide by a fraction, we multiply by its reciprocal: $ S = 4 \cdot \frac{3}{2} $.Perform multiplication: We perform the multiplication: $ S = 4 \cdot \frac{3}{2} = 2 \cdot 3 = 6 $.

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$\sum_{k=1}^{\infty} 4\left(\frac{1}{3}\right)^{k-1}$

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Algebra 2

Sum of finite series starts from 1

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\sum_{k=1}^{\infty} 4\left(\frac{1}{3}\right)^{k-1}

Recognize series type: We recognize that the series is a geometric series with the first term $a = 4$ (since when $k = 1$ , the term is $4 \cdot 1$ ) and the common ratio $r = \frac{1}{3}$ . The sum of an infinite geometric series can be found using the formula $S = \frac{a}{1 - r}$ , where |r| < 1 for the series to converge.
Check convergence: We check the common ratio to ensure that the series converges. Since |r| = |\frac{1}{3}| = \frac{1}{3} < 1 , the series converges.
Apply sum formula: We apply the formula for the sum of an infinite geometric series: $S = \frac{a}{1 - r}$ . Substituting $a = 4$ and $r = \frac{1}{3}$ , we get $S = \frac{4}{1 - \frac{1}{3}}$ .
Simplify denominator: We simplify the denominator: $1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}$ .
Calculate sum: We calculate the sum: $S = \frac{4}{\frac{2}{3}}$ . To divide by a fraction, we multiply by its reciprocal: $S = 4 \cdot \frac{3}{2}$ .
Perform multiplication: We perform the multiplication: $S = 4 \cdot \frac{3}{2} = 2 \cdot 3 = 6$ .