Determine if the Infinite Series SUM( 1/(sqrt(2))^n)) Converge or Diverges  Summary and Q&A
TL;DR
Determine if an infinite series converges or diverges and find its sum.
Key Insights
 🍹 The provided problem involves determining the convergence or divergence of an infinite series and finding its sum.
 ✊ Geometric series follow the form a times r raised to the power of n.
 🥳 The geometric series test is a method used to identify the convergence or divergence of a series based on the absolute value of the common ratio.
 🥳 In this case, the common ratio is 1 divided by the square root of 2, and its absolute value is less than one, indicating convergence.
 🥳 The sum of a geometric series can be found by using the formula a / (1  r), where a is the first term and r is the common ratio.
 🥳 To find the sum in this problem, the first term 1 / √2 is divided by 1 minus the common ratio (1  1 / √2).
Questions & Answers
Q: What is the geometric series test?
The geometric series test states that if the absolute value of the common ratio (r) in a geometric series is less than one, the series converges. If it is greater than or equal to one, the series diverges.
Q: How is the common ratio (r) determined in this problem?
In this problem, the common ratio is 1 divided by the square root of 2. This value is obtained by analyzing the given series.
Q: How do you find the sum of a geometric series?
To find the sum of a geometric series, you plug the common ratio into the formula: a / (1  r), where a is the first term and r is the common ratio. In this case, the sum is calculated as 1 / (√2  1 / √2).
Q: Can you explain the process of finding the sum in more detail?
To find the sum, you take the first term (1 / √2) and divide it by 1 minus the common ratio (1  1 / √2). Simplifying the expression yields the sum as (√2 / √2) / ((√2  1) / √2), which further simplifies to √2 / (√2  1).
Summary & Key Takeaways

The problem involves determining whether an infinite series converges or diverges and finding the sum.

The series is identified as a geometric series, which follows the pattern a times r raised to the power of n.

Applying the geometric series test, it is found that the series converges as the absolute value of the common ratio is less than one.