sum_(k=1)^(oo)(sin 100)^(k)

Recognize Geometric Series: We recognize that the series is a geometric series with the first term a = sin(100) and the common ratio r = sin(100). The sum of an infinite geometric series can be found using the formula S = a / (1 - r), where |r| < 1. Check Common Ratio: First, we need to check if the common ratio r = sin(100) satisfies the condition |r| < 1. Since the sine function always returns a value between -1 and 1 for any angle, we have |sin(100)| < 1. Apply Formula for Sum: Now we can apply the formula for the sum of an infinite geometric series: S = a / (1 - r). Here, a = sin(100) and r = sin(100), so S = sin(100) / (1 - sin(100)). Perform Calculation: We perform the calculation: S = sin(100) / (1 - sin(100)). Note that sin(100) is in degrees, and we should convert it to radians if we are using a calculator that requires radians. However, since we are not actually calculating the numerical value of sin(100), we can leave it as is. Final Answer: The final answer is S = sin(100) / (1 - sin(100)). This is the sum of the infinite series.

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Algebra 2

Sum of finite series starts from 1

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\sum_{k=1}^{\infty}(\sin 100)^{k}

Recognize Geometric Series: We recognize that the series is a geometric series with the first term $a = \sin(100)$ and the common ratio $r = \sin(100)$ . The sum of an infinite geometric series can be found using the formula $S = \frac{a}{1 - r}$ , where |r| < 1.
Check Common Ratio: First, we need to check if the common ratio $r = \sin(100)$ satisfies the condition |r| < 1. Since the sine function always returns a value between $-1$ and $1$ for any angle, we have |\sin(100)| < 1.
Apply Formula for Sum: Now we can apply the formula for the sum of an infinite geometric series: $S = \frac{a}{1 - r}$ . Here, $a = \sin(100)$ and $r = \sin(100)$ , so $S = \frac{\sin(100)}{1 - \sin(100)}$ .
Perform Calculation: We perform the calculation: $S = \frac{\sin(100)}{1 - \sin(100)}$ . Note that $\sin(100)$ is in degrees, and we should convert it to radians if we are using a calculator that requires radians. However, since we are not actually calculating the numerical value of $\sin(100)$ , we can leave it as is.
Final Answer: The final answer is $S = \frac{\sin(100)}{1 - \sin(100)}$ . This is the sum of the infinite series.