An infinite series is a sum of an infinite number of terms. It can be written in the form of a1 + a2 + a3 + ... + an, where a1, a2, a3, etc. are the terms of the series. The value of an infinite series depends on the individual values of each term and the pattern in which they are added together.
To determine if an infinite series converges or diverges, you can use various tests such as the ratio test, the comparison test, or the integral test. These tests evaluate the behavior of the terms in the series and determine if the series approaches a finite value (converges) or approaches infinity (diverges).
A convergent series is one that approaches a finite value as more terms are added, while a divergent series is one that approaches infinity (or negative infinity) as more terms are added. In other words, a convergent series has a finite sum, while a divergent series does not.
Yes, an infinite series can converge to a negative value. This means that the sum of the series will be a negative number. For example, the infinite series 1 - 1 + 1 - 1 + ... converges to the value of 1/2, which is a negative number.
The limit comparison test compares the behavior of the terms in the given series to those in a known series whose convergence or divergence is already known. To use this test, you take the limit of the ratio of the terms in the given series and the known series. If the limit is a positive finite value, then the two series have the same behavior and will either both converge or both diverge. If the limit is zero or infinity, then the two series have different behavior and the convergence of the given series cannot be determined using this test.