Does the Series Converge or Diverge? SUM( cos(1)^k ) Example with the Geometric Series Test | Summary and Q&A

TL;DR

The video explains how to determine the convergence and find the sum of an infinite geometric series with cosine.

Key Insights

• ✊ The series is determined to be a geometric series because it follows the form of a number (a) times the common ratio (r) raised to the power of n.
• 🥳 The common ratio in the series is found to be the cosine of one, which is approximately 0.54.
• 🥳 By applying the geometric series test, it is concluded that the series converges because the absolute value of the common ratio is less than one.

Transcript

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Questions & Answers

Q: How can we identify if a series converges or diverges using the geometric series test?

The geometric series test states that if the absolute value of the common ratio (r) is less than one, the series converges. If it is greater than or equal to one, the series diverges.

Q: What is the common ratio in the given series?

The common ratio in the given series is the cosine of one.

Q: How can we determine that the cosine of one is less than one?

Graphically, we can observe that the value of cosine of one lies between the values of one and zero on the graph of the cosine function, indicating that it is less than one numerically.

Q: How can we find the sum of the given infinite series?

To find the sum, we plug the common ratio (cosine of one) into the formula: cosine of 1 over 1 minus cosine of 1.

Summary & Key Takeaways

• The video discusses an infinite series and confirms it is a geometric series with a common ratio of the cosine of one.

• Using the geometric series test, it is determined that the series converges because the absolute value of the common ratio is less than one.

• The sum of the series is found by plugging the common ratio (cosine of one) into the formula: cosine of 1 over 1 minus cosine of 1.