Find the infinite sum of the geometric sequence with a = 6, r 2 if it exists.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Problem: Infinite Sum of a Geometric Sequence**
Find the infinite sum of the geometric sequence with the first term \( a = 6 \) and common ratio \( r = \frac{2}{6} \) if it exists.
\[ S_\infty = \boxed{\phantom{text}} \]
**Explanation:**
For a geometric sequence, the infinite sum \( S_\infty \) exists if the absolute value of the common ratio \( r \) is less than 1. The formula for the infinite sum is:
\[ S_\infty = \frac{a}{1 - r} \]
Substitute the given values:
- \( a = 6 \)
- \( r = \frac{2}{6} = \frac{1}{3} \)
Calculate \( S_\infty \) using the formula.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd920f800-7cb4-4708-a86f-fdc64006663e%2F21a81df9-b28b-4b59-a900-b6f5dedd7e0a%2Fg83fq6a_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Infinite Sum of a Geometric Sequence**
Find the infinite sum of the geometric sequence with the first term \( a = 6 \) and common ratio \( r = \frac{2}{6} \) if it exists.
\[ S_\infty = \boxed{\phantom{text}} \]
**Explanation:**
For a geometric sequence, the infinite sum \( S_\infty \) exists if the absolute value of the common ratio \( r \) is less than 1. The formula for the infinite sum is:
\[ S_\infty = \frac{a}{1 - r} \]
Substitute the given values:
- \( a = 6 \)
- \( r = \frac{2}{6} = \frac{1}{3} \)
Calculate \( S_\infty \) using the formula.
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