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Home » The sum of the first n terms of a geometric series is \( S_n = a \fracr^n - 1r - 1 \). - AMAZONAWS

The sum of the first n terms of a geometric series is \( S_n = a \fracr^n - 1r - 1 \). - AMAZONAWS

📅 March 6, 2026 👤 scraface

Mar 06, 2026

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However, to **maximize $r$**, we must **minimize the denominator** $\s...
Actually, the **maximum finite value of $r$** occurs when $\sin(2\thet...
r^2 \leq \frac{100}{1} = 100 \Rightarrow r \leq 10.
But we must check if this maximum is **attainable**. That is, does the...
Set $\sin(2\theta) = 1 \Rightarrow 2\theta = \frac{\pi}{2} + 2k\pi \Ri...
Try $\theta = \frac{\pi}{4}$:
Then $r^2 = \frac{100}{1} = 100 \Rightarrow r = 10$.
Check if this point lies within the circular region:
The circle has radius 10, so $r = 10$ is on the boundary — valid.
Thus, the **maximum possible value of $r$** is:

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