Determine whether the following series converges. Justify your answer.The convergence of infinite series is a fundamental concept in mathematics, with far-reaching applications in various fields. This comprehensive guide delves into the topic, exploring the convergence tests, their applications, and the justification behind their effectiveness.
We will delve into the intricacies of the ratio test, root test, integral test, comparison test, limit comparison test, and alternating series test. Each test provides a unique perspective on series convergence, empowering us to make informed judgments about their behavior.
Convergence Tests for Infinite Series: Determine Whether The Following Series Converges. Justify Your Answer.
In mathematics, a convergence test is a tool used to determine whether an infinite series converges, meaning that its sum approaches a finite limit as the number of terms in the series increases indefinitely.
Ratio Test
The ratio test is a convergence test that uses the ratio of consecutive terms in the series to determine convergence. If the limit of the ratio is less than 1, the series converges; if the limit is greater than 1, the series diverges; and if the limit is equal to 1, the test is inconclusive.
Root Test
The root test is a convergence test that uses the nth root of the absolute value of the nth term in the series to determine convergence. If the limit of the nth root is less than 1, the series converges; if the limit is greater than 1, the series diverges; and if the limit is equal to 1, the test is inconclusive.
Integral Test
The integral test is a convergence test that uses an integral to determine the convergence of a series. If the integral of the corresponding function is convergent, then the series is also convergent; if the integral is divergent, then the series is also divergent.
Comparison Test, Determine whether the following series converges. justify your answer.
The comparison test is a convergence test that compares the given series to a known convergent or divergent series. If the given series is less than or equal to the convergent series, then the given series is also convergent; if the given series is greater than or equal to the divergent series, then the given series is also divergent.
Limit Comparison Test
The limit comparison test is a convergence test that compares the given series to a known convergent or divergent series by taking the limit of the ratio of the two series. If the limit is finite and nonzero, then the given series converges if and only if the known series converges.
Alternating Series Test
The alternating series test is a convergence test that applies to series with alternating signs. If the absolute values of the terms decrease monotonically and approach zero, then the series converges.
Detailed FAQs
What is the ratio test?
The ratio test is a convergence test that examines the limit of the ratio of consecutive terms in a series. If the limit is less than 1, the series converges absolutely. If the limit is greater than 1, the series diverges.
If the limit is equal to 1, the ratio test is inconclusive.
How do I apply the root test to determine convergence?
The root test involves taking the nth root of the absolute value of the nth term in a series and finding its limit. If the limit is less than 1, the series converges absolutely. If the limit is greater than 1, the series diverges.
If the limit is equal to 1, the root test is inconclusive.
