Question

1. does the infinite geometric series \\(s = 25 + 20 + 16 + \dots\\) converge or diverge? explain.

• the infinite geometric series converges because the value of the first term in the series is less than 50.
• the common ratio is 1.25, which is not between the values of -1 and 1 so the infinite geometric series diverges.
• the common ratio is 0.80, which is between the values of -1 and 1 so the infinite geometric series converges.
• the infinite geometric series diverges because you do not know the values of the terms in the series that follow the

Explanation:

🆕 New Concept Discovered: Infinite Geometric Series
How infinite sums settle on a single value

Step 1: Find the common ratio

To find the common ratio \( r \), divide the second term by the first term:

$$ r = \frac{20}{25} = 0.80 $$

Verify with the next term:

$$ \frac{16}{20} = 0.80 $$

Step 2: Determine convergence

An infinite geometric series converges if and only if the absolute value of its common ratio is strictly less than 1:

$$ |r| < 1 \implies -1 < r < 1 $$

Since \( r = 0.80 \), it lies between \(-1\) and \(1\). Therefore, the series converges.

Answer:

The common ratio is 0.80, which is between the values of -1 and 1 so the infinite geometric series converges.