sum_(n=1)^(oo)3^(n+1)4^(-n)

Recognize as geometric series: Recognize the series as a geometric series. A geometric series has the form $ \sum_{n=0}^{\infty} ar^n $, where $ a $ is the first term and $ r $ is the common ratio.Identify first term and ratio: Identify the first term $ a $ and the common ratio $ r $ of the series. The first term when $ n = 1 $ is $ 3^{1+1} 4^{-1} = 3^2 / 4 = 9/4 $. The common ratio $ r $ is obtained by dividing the term for $ n+1 $ by the term for $ n $, which gives $ (3^{(n+1)+1} 4^{-(n+1)}) / (3^{n+1} 4^{-n}) = 3/4 $.Check convergence condition: Check if the common ratio $ r $ is between -1 and 1, which is a necessary condition for the convergence of a geometric series. Since $ r = 3/4 $, which is between -1 and 1, the series converges.Use sum formula: Use the formula for the sum of an infinite geometric series, which is $ S = a / (1 - r) $, where $ S $ is the sum, $ a $ is the first term, and $ r $ is the common ratio. Substitute $ a = 9/4 $ and $ r = 3/4 $ into the formula.Calculate sum: Calculate the sum using the formula $ S = 9/4 / (1 - 3/4) $. Simplify the denominator to get $ S = 9/4 / (1/4) $.Simplify expression: Simplify the expression to find the sum. $ S = 9/4 * 4/1 = 9 $.

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Algebra 2

Sum of finite series starts from 1

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\sum_{n=1}^{\infty}3^{n+1}4^{-n}

Recognize as geometric series: Recognize the series as a geometric series. A geometric series has the form $\sum_{n=0}^{\infty} ar^n$ , where $a$ is the first term and $r$ is the common ratio.
Identify first term and ratio: Identify the first term $a$ and the common ratio $r$ of the series. The first term when $n = 1$ is $3^{1+1} 4^{-1} = 3^2 / 4 = 9/4$ . The common ratio $r$ is obtained by dividing the term for $n+1$ by the term for $n$ , which gives $(3^{(n+1)+1} 4^{-(n+1)}) / (3^{n+1} 4^{-n}) = 3/4$ .
Check convergence condition: Check if the common ratio $r$ is between $-1$ and $1$ , which is a necessary condition for the convergence of a geometric series. Since $r = 3/4$ , which is between $-1$ and $1$ , the series converges.
Use sum formula: Use the formula for the sum of an infinite geometric series, which is $S = a / (1 - r)$ , where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio. Substitute $a = 9/4$ and $r = 3/4$ into the formula.
Calculate sum: Calculate the sum using the formula $S = 9/4 / (1 - 3/4)$ . Simplify the denominator to get $S = 9/4 / (1/4)$ .
Simplify expression: Simplify the expression to find the sum. $S = 9/4 * 4/1 = 9$ .