Determine if the Infinite Series SUM( 1/(sqrt(2))^n)) Converge or Diverges

TL;DR

Determine if an infinite series converges or diverges and find its sum.

Transcript

in this problem we have an infinite series and the question is determine if it converges or diverges and if it converges uh find the sum so this appears to be a geometric series so geometric series generally have this form you know a times r to the n or some books will will write you know a times r to the n minus one the thing is they basically hav... Read More

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.