A convergent series is a sequence of numbers that approaches a finite limit as the number of terms increases. This means that the sum of the terms in the series gets closer and closer to a specific value as more terms are added.
To find the sequence of positive real numbers for a convergent series, you can use the formula for the general term of the series. This formula will give you a pattern to follow for finding the individual terms of the sequence.
No, a convergent series cannot have negative terms. This is because a negative term would cause the sum of the series to become smaller as more terms are added, making it diverge rather than converge.
There are several tests that can be used to determine if a series is convergent or divergent, such as the ratio test, the comparison test, and the integral test. These tests involve analyzing the behavior of the terms in the series and can help determine if the series will approach a finite limit or not.
Convergent series are used in many scientific and mathematical applications, such as calculating the value of pi, approximating the area under a curve, and predicting population growth. They are also commonly used in physics and engineering to model real-world systems and make accurate predictions.