Question

1. which statement about convergent infinite geometric series is true?

a finite geometric sequence will have an infinite geometric series.
the graph of a convergent infinite geometric series goes to infinity.
an infinite geometric series converges if \\(|r| < 1\\) is true.
the graph of an infinite geometric series curves away from its sum.

Explanation:

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

Step 1: Understand convergence of infinite geometric series

An infinite geometric series has the general form:

$$ \sum_{n=1}^{\infty} a_1 r^{n-1} = a_1 + a_1 r + a_1 r^2 + a_1 r^3 + \dots $$

For the sum of infinitely many terms to approach a single, finite number (to converge), the terms must get smaller and smaller, shrinking toward zero. This happens if and only if the common ratio \( r \) lies strictly between \(-1\) and \(1\):

$$ |r| < 1 \quad \text{or} \quad -1 < r < 1 $$

Step 2: Evaluate the given options

• Option 1: "A finite geometric sequence will have an infinite geometric series."
• This is false. A finite sequence has a limited number of terms, so its corresponding series is also finite, not infinite.

• Option 2: "The graph of a convergent infinite geometric series goes to infinity."
• This is false. If a series converges, its sum approaches a specific finite value (a horizontal asymptote), not infinity.

• Option 3: "An infinite geometric series converges if \( |r| < 1 \) is true."
• This is true. The highlighted box in the image represents \( |r| < 1 \), which is the exact mathematical condition required for convergence.

• Option 4: "The graph of an infinite geometric series curves away from its sum."
• This is false. The graph of the partial sums of a convergent series curves toward (approaches) its sum.

Answer:

An infinite geometric series converges if \( |r| < 1 \) is true.