Calculus 2  Geometric Series, PSeries, Ratio Test, Root Test, Alternating Series, Integral Test  Summary and Q&A
TL;DR
Learn about different tests to determine if a series converges or diverges, including the divergence test, geometric series test, pseries test, telescoping series test, integral test, ratio test, root test, direct comparison test, and limit comparison test.
Questions & Answers
Q: What is the divergence test?
The divergence test is a method to determine if a series diverges. It involves taking the limit as n approaches infinity of the sequence. If the limit is not zero, the series diverges.
Q: How does the geometric series test work?
The geometric series test determines if a series converges by identifying the common ratio. If the absolute value of the common ratio is less than one, the series converges. If it is greater than or equal to one, the series diverges.
Q: What is the pseries test?
The pseries test determines if a series converges based on the exponent of 1/n. If the exponent is greater than 1, the series converges. If it is less than or equal to 1, the series diverges.
Q: How does the telescoping series test work?
The telescoping series test involves cancelling terms in a series to simplify it. Then, the sum of the series is evaluated as n approaches infinity. If the sum is a finite value, the series converges. If it is infinity or doesn't exist, the series diverges.
Q: What is the integral test?
The integral test compares a series to the integral of a function. If the integral converges, the series converges. If it diverges, the series diverges.
Q: How does the ratio test work?
The ratio test compares the absolute values of consecutive terms in a series. If the limit of the ratio is less than 1, the series converges. If it is greater than or equal to 1, the series diverges.
Q: What is the root test?
The root test determines convergence or divergence by taking the limit of the nth root of the absolute value of the terms. If the limit is less than 1, the series converges. If it is greater than or equal to 1, the series diverges.
Q: How do the direct comparison and limit comparison tests work?
The direct comparison test compares a series to another series. If the big series converges, the small series converges as well, and if the small series diverges, the big series diverges. The limit comparison test involves taking the limit of the ratio of terms in two series to determine their convergence or divergence.
Summary & Key Takeaways

The divergence test determines if a series diverges by taking the limit as n approaches infinity of the sequence. If the limit is not zero, the series diverges.

The geometric series test determines if a series converges by identifying the common ratio. If the absolute value of the common ratio is less than one, the series converges. If it is greater than or equal to one, the series diverges.

The pseries test determines if a series converges based on the exponent of 1/n. If the exponent is greater than 1, the series converges. If it is less than or equal to 1, the series diverges.

The telescoping series test involves cancelling terms to simplify the series and then evaluating the sum as n approaches infinity. If the sum is a finite value, the series converges. If it is infinity or doesn't exist, the series diverges.

The integral test involves comparing a series to the integral of a function. If the integral converges, the series converges. If it diverges, the series diverges.

The ratio test compares the absolute values of consecutive terms in a series to determine convergence or divergence. If the limit is less than 1, the series converges. If it is greater than or equal to 1, the series diverges.

The root test determines convergence or divergence by taking the limit of the nth root of the absolute value of the terms. If the limit is less than 1, the series converges. If it is greater than or equal to 1, the series diverges.

The direct comparison and limit comparison tests involve comparing a series to another series. If the big series converges, the small series converges as well, and if the small series diverges, the big series diverges.

The alternating series test is used for series with alternating signs. It checks conditions of divergence, decreasing sequence, and absolute convergence to determine convergence or divergence.