Alternating Series An alternating series is a sequence of numbers where the signs of the terms change. This type of series can converge or diverge depending...
Alternating Series
An alternating series is a sequence of numbers where the signs of the terms change. This type of series can converge or diverge depending on the behavior of its individual terms.
Convergence and Divergence:
An alternating series converges if the sequence converges to a finite number.
An alternating series diverges if the sequence approaches infinity.
Convergence Tests:
To determine if an alternating series converges, we can use various tests, including:
The Ratio Test: If the limit of the absolute value of the ratio between two consecutive terms is equal to a finite number, the series converges.
The Root Test: If the limit of the square root of the absolute value of the difference between two consecutive terms is equal to a finite number, the series converges.
The Alternating Series Test: If the sequence follows the pattern of an alternating series with a common difference, it converges.
Examples:
The series (\sum\limits_{n=1}^\infty (-1)^n) converges because the limit of the ratio of consecutive terms is -1, which is less than 1.
The series (\sum\limits_{n=1}^\infty \frac{1}{n}) diverges because the limit of the absolute value of the ratio of consecutive terms is infinity.
The series (\sum\limits_{n=1}^\infty \frac{(-1)^n}{n}) converges because the sequence alternates between positive and negative signs