Absolute Convergence: An infinite series converges absolutely if the series' sum converges. This means that the limit of the sequence as n approaches infini...
Absolute Convergence:
An infinite series converges absolutely if the series' sum converges. This means that the limit of the sequence as n approaches infinity is finite and non-zero. The series converges absolutely if the sum exists and is finite.
Examples:
The series converges absolutely: ∑(-1)^n converges because the limit of the sequence as n approaches infinity is 0.
The series diverges absolutely: ∑1/n converges because the limit of the sequence as n approaches infinity is infinity.
Conditional Convergence:
An infinite series converges conditionally if the series converges if n is greater than a certain value and diverges if n is less than that value. This means that the series converges if the sequence oscillates between two limits. The series converges conditionally if the limit as n approaches infinity exists but is different from both positive and negative infinity.
Examples:
The series converges conditionally: ∑(1/n) converges because it oscillates between 0 and 1 for n > 1.
The series diverges conditionally: ∑(-1)^n converges because it oscillates between -1 and 1 for n > 1.
Understanding absolute and conditional convergence is crucial for analyzing the convergence behavior of infinite series. It helps to determine whether a series converges or diverges and provides insight into the behavior of the series in different intervals