cacophony said:1. $\displaystyle \sum_{n=0}^{\infty} \frac{n!}{n! + 3^n}$
cacophony said:2.
infinity
E...(n - (1/n))^-n
n = 1
An infinite series is a mathematical expression that consists of an infinite sum of terms. It can be written as a sum of terms starting from the first term and continuing indefinitely, each term being added to the previous one.
The convergence or divergence of an infinite series can be determined by using various tests such as the ratio test, the root test, and the comparison test. These tests evaluate the behavior of the terms in the series and determine if they approach a finite limit or if they continue to increase without bound.
A convergent series is an infinite series that has a finite limit. This means that the sum of the terms in the series approaches a finite value as the number of terms increases. In other words, the series "converges" to a specific value.
A divergent series is an infinite series that does not approach a finite limit. This means that the sum of the terms in the series increases without bound as the number of terms increases. In other words, the series "diverges" and does not have a specific value.
Understanding infinite series convergence is important in various fields of science and mathematics, such as physics, engineering, and statistics. It allows us to accurately calculate and predict the behavior of complex systems and phenomena. Additionally, it is a fundamental concept in calculus and is necessary for solving many mathematical problems.