Check Series For Convergence

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Introduction

In mathematics, a series is a sequence of numbers that are added together to form a sum. The study of series is crucial in various fields, including calculus, analysis, and number theory. One of the fundamental concepts in series is convergence, which refers to the behavior of a series as the number of terms increases without bound. In this article, we will discuss how to check series for convergence, with a focus on the given series: $ \sum_{n = 1}^\infty \sin(n)\sin\left(\frac{(-1)n}{n{1/4}}\right) $

Understanding Convergence

Convergence of a series is determined by the behavior of its terms as the number of terms increases without bound. A series is said to converge if the sum of its terms approaches a finite limit as the number of terms approaches infinity. On the other hand, a series is said to diverge if the sum of its terms does not approach a finite limit as the number of terms approaches infinity.

Types of Convergence

There are several types of convergence, including:

  • Pointwise convergence: A series converges pointwise if the sequence of partial sums converges to a limit for each individual term.
  • Uniform convergence: A series converges uniformly if the sequence of partial sums converges to a limit at the same rate for all terms.
  • Absolute convergence: A series converges absolutely if the series of absolute values of its terms converges.

Checking Convergence

To check convergence of a series, we can use various tests, including:

  • Ratio test: This test involves comparing the ratio of consecutive terms to determine if the series converges.
  • Root test: This test involves comparing the nth root of the nth term to determine if the series converges.
  • Integral test: This test involves comparing the integral of the function to determine if the series converges.
  • Comparison test: This test involves comparing the series to a known convergent or divergent series to determine if the series converges.

The Given Series

The given series is: $ \sum_{n = 1}^\infty \sin(n)\sin\left(\frac{(-1)n}{n{1/4}}\right) $

This series involves the product of two trigonometric functions: sine and sine. The first term involves the sine function with a periodic argument, while the second term involves the sine function with a rapidly decreasing argument.

Applying the Ratio Test

To check convergence of the given series, we can apply the ratio test. The ratio test involves comparing the ratio of consecutive terms to determine if the series converges.

Let's define the nth term of the series as: $ a_n = \sin(n)\sin\left(\frac{(-1)n}{n{1/4}}\right) $

Then, the ratio of consecutive terms is: $ \frac{a_{n+1}}{a_n} = \frac{\sin(n+1)\sin\left(\frac{(-1){n+1}}{(n+1){1/4}}\right)}{\sin(n)\sin\left(\frac{(-1)n}{n{14}}\right)} $

Using the product-to-sum formula for sine, we can simplify the ratio of consecutive terms as: $ \frac{a_{n+1}}{a_n} = \frac{\sin(n+1)\sin\left(\frac{(-1){n+1}}{(n+1){1/4}}\right)}{\sin(n)\sin\left(\frac{(-1)n}{n{1/4}}\right)} = \frac{\cos\left(n+\frac{1}{2}\right)\sin\left(\frac{(-1){n+1}}{(n+1){1/4}}\right)}{\sin(n)\sin\left(\frac{(-1)n}{n{1/4}}\right)} $

Using the identity cos⁡(n+12)=−cos⁡(n−12)\cos\left(n+\frac{1}{2}\right) = -\cos\left(n-\frac{1}{2}\right), we can simplify the ratio of consecutive terms as: $ \frac{a_{n+1}}{a_n} = \frac{-\cos\left(n-\frac{1}{2}\right)\sin\left(\frac{(-1){n+1}}{(n+1){1/4}}\right)}{\sin(n)\sin\left(\frac{(-1)n}{n{1/4}}\right)} $

Now, we can take the limit of the ratio of consecutive terms as n approaches infinity: $ \lim_{n\to\infty} \frac{a_{n+1}}{a_n} = \lim_{n\to\infty} \frac{-\cos\left(n-\frac{1}{2}\right)\sin\left(\frac{(-1){n+1}}{(n+1){1/4}}\right)}{\sin(n)\sin\left(\frac{(-1)n}{n{1/4}}\right)} $

Using the fact that cos⁡(n−12)\cos\left(n-\frac{1}{2}\right) approaches 0 as n approaches infinity, we can simplify the limit of the ratio of consecutive terms as: $ \lim_{n\to\infty} \frac{a_{n+1}}{a_n} = \lim_{n\to\infty} \frac{-\sin\left(\frac{(-1){n+1}}{(n+1){1/4}}\right)}{\sin(n)\sin\left(\frac{(-1)n}{n{1/4}}\right)} $

Using the fact that sin⁡((−1)n+1(n+1)1/4)\sin\left(\frac{(-1)^{n+1}}{(n+1)^{1/4}}\right) approaches 0 as n approaches infinity, we can simplify the limit of the ratio of consecutive terms as: $ \lim_{n\to\infty} \frac{a_{n+1}}{a_n} = \lim_{n\to\infty} \frac{-0}{\sin(n)\sin\left(\frac{(-1)n}{n{1/4}}\right)} = 0 $

Since the limit of the ratio of consecutive terms is 0, the series converges by the ratio test.

Conclusion

In this article, we discussed how to check series for convergence, with a focus on the given series: $ \sum_{n = 1}^\infty \sin(n)\sin\left(\frac{(-1)n}{n{1/4}}right) $

We applied the ratio test to check convergence of the series and found that the series converges. The ratio test is a powerful tool for checking convergence of series, and it can be used to determine if a series converges or diverges.

References

  • Walter Rudin, "Principles of Mathematical Analysis", McGraw-Hill, 1976.
  • George F. Simmons, "Calculus with Analytic Geometry", McGraw-Hill, 1985.
  • James Stewart, "Calculus: Early Transcendentals", Brooks Cole, 2007.

Further Reading

  • Convergence Tests: A comprehensive guide to convergence tests, including the ratio test, root test, integral test, and comparison test.
  • Series and Sequences: A detailed discussion of series and sequences, including definitions, properties, and examples.
  • Calculus: A comprehensive guide to calculus, including limits, derivatives, integrals, and applications.
    Check Series for Convergence: A Comprehensive Guide ===========================================================

Q&A: Check Series for Convergence

Q: What is a series?

A: A series is a sequence of numbers that are added together to form a sum.

Q: What is convergence?

A: Convergence refers to the behavior of a series as the number of terms increases without bound. A series is said to converge if the sum of its terms approaches a finite limit as the number of terms approaches infinity.

Q: How do I check if a series converges?

A: There are several tests that can be used to check if a series converges, including the ratio test, root test, integral test, and comparison test.

Q: What is the ratio test?

A: The ratio test involves comparing the ratio of consecutive terms to determine if the series converges. If the limit of the ratio of consecutive terms is less than 1, the series converges.

Q: What is the root test?

A: The root test involves comparing the nth root of the nth term to determine if the series converges. If the limit of the nth root of the nth term is less than 1, the series converges.

Q: What is the integral test?

A: The integral test involves comparing the integral of the function to determine if the series converges. If the integral of the function is finite, the series converges.

Q: What is the comparison test?

A: The comparison test involves comparing the series to a known convergent or divergent series to determine if the series converges.

Q: How do I apply the ratio test?

A: To apply the ratio test, you need to calculate the ratio of consecutive terms and take the limit as n approaches infinity. If the limit is less than 1, the series converges.

Q: How do I apply the root test?

A: To apply the root test, you need to calculate the nth root of the nth term and take the limit as n approaches infinity. If the limit is less than 1, the series converges.

Q: How do I apply the integral test?

A: To apply the integral test, you need to calculate the integral of the function and determine if it is finite. If the integral is finite, the series converges.

Q: How do I apply the comparison test?

A: To apply the comparison test, you need to compare the series to a known convergent or divergent series. If the series is less than or equal to the convergent series, the series converges.

Q: What are some common mistakes to avoid when checking series for convergence?

A: Some common mistakes to avoid when checking series for convergence include:

  • Not using the correct test for the series
  • Not taking the limit correctly
  • Not checking for convergence at the endpoints
  • Not considering the possibility of conditional convergence

Q: What are some tips for checking series for convergence?

A: Some tips for checking series for convergence include:

  • Start by checking if the series is a geometric series or an arithmetic
  • Use the ratio test or root test for series with terms that involve powers of n
  • Use the integral test for series with terms that involve integrals
  • Use the comparison test for series that are similar to known convergent or divergent series
  • Check for convergence at the endpoints of the series

Conclusion

In this article, we discussed how to check series for convergence, with a focus on the given series: $ \sum_{n = 1}^\infty \sin(n)\sin\left(\frac{(-1)n}{n{1/4}}\right) $

We also provided a Q&A section to answer common questions about checking series for convergence. By following the tips and avoiding common mistakes, you can effectively check series for convergence and determine if they converge or diverge.

References

  • Walter Rudin, "Principles of Mathematical Analysis", McGraw-Hill, 1976.
  • George F. Simmons, "Calculus with Analytic Geometry", McGraw-Hill, 1985.
  • James Stewart, "Calculus: Early Transcendentals", Brooks Cole, 2007.

Further Reading

  • Convergence Tests: A comprehensive guide to convergence tests, including the ratio test, root test, integral test, and comparison test.
  • Series and Sequences: A detailed discussion of series and sequences, including definitions, properties, and examples.
  • Calculus: A comprehensive guide to calculus, including limits, derivatives, integrals, and applications.