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  2. AP Calculus
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How to determine if the series ∑n=0∞5cdot(23)n\sum_{n=0}^{\infty} 5 cdot (\frac{2}{3})^n∑n=0∞​5cdot(32​)n converges or diverges?

Identify 'a' (5) and 'r' (2/3). Since |2/3| < 1, the series converges by the geometric series test.

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How to determine if the series ∑n=0∞5cdot(23)n\sum_{n=0}^{\infty} 5 cdot (\frac{2}{3})^n∑n=0∞​5cdot(32​)n converges or diverges?

Identify 'a' (5) and 'r' (2/3). Since |2/3| < 1, the series converges by the geometric series test.

How to find the sum of the series ∑n=1∞4cdot(12)n−1\sum_{n=1}^{\infty} 4 cdot (\frac{1}{2})^{n-1}∑n=1∞​4cdot(21​)n−1?

Identify 'a' (4) and 'r' (1/2). Use the formula a/(1-r) = 4/(1-1/2) = 8. The sum is 8.

How to determine if the series ∑n=0∞3cdot(−2)n\sum_{n=0}^{\infty} 3 cdot (-2)^n∑n=0∞​3cdot(−2)n converges or diverges?

Identify 'a' (3) and 'r' (-2). Since |-2| ≥ 1, the series diverges by the geometric series test.

Given the sequence 4, 1, 1/4, 1/16,... how do you write the geometric series and determine its convergence?

a = 4, r = 1/4. The series is ∑n=0∞4cdot(14)n\sum_{n=0}^{\infty} 4 cdot (\frac{1}{4})^n∑n=0∞​4cdot(41​)n. Since |1/4| < 1, the series converges.

How do you find the sum of the infinite geometric series: 1 + 0.1 + 0.01 + 0.001 + ...?

Recognize a = 1, r = 0.1. Sum = a / (1 - r) = 1 / (1 - 0.1) = 1 / 0.9 = 10/9.

General form of a geometric series (starting from n=0)?

∑n=0∞acdotrn\sum_{n=0}^{\infty} a cdot r^nn=0∑∞​acdotrn

General form of a geometric series (starting from n=1)?

∑n=1∞acdotrn−1\sum_{n=1}^{\infty} a cdot r^{n-1}n=1∑∞​acdotrn−1

Formula for the sum of a converging geometric series?

∑n=1∞acdotrn−1=∑n=0∞acdotrn=a1−r\sum_{n=1}^{\infty} acdot r^{n-1} = \sum_{n=0}^{\infty} a cdot r^n = \frac{a}{1-r}n=1∑∞​acdotrn−1=n=0∑∞​acdotrn=1−ra​

Explain the significance of the common ratio 'r' in determining convergence.

The absolute value of 'r' determines convergence/divergence: if 0 < |r| < 1, the series converges; if |r| ≥ 1, the series diverges.

How does changing the starting index of a geometric series affect its convergence?

Changing the starting index (e.g., from n=0 to n=1) does not affect whether the series converges or diverges.

What is the key idea behind the geometric series test?

The geometric series test provides a straightforward condition based on the common ratio 'r' to determine if an infinite geometric series converges or diverges.