mathman said:Did you try to get the derivation of the partial sum formula? Once you have that the final result is obvious.
A simple convergent series is a mathematical series where the terms of the series approach a finite limit as the number of terms increases. In other words, the sum of the series gets closer and closer to a specific value as more terms are added.
A divergent series is a mathematical series where the terms do not approach a finite limit as the number of terms increases. Instead, the sum of the series either gets larger and larger or oscillates between different values as more terms are added. This means that a divergent series does not have a specific sum or limit.
The formula for a simple convergent series is: S = a + ar + ar^2 + ar^3 + ... + ar^n, where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms in the series. This formula only applies to geometric series, which are a type of simple convergent series.
A simple convergent series is convergent if the common ratio (r) is between -1 and 1. If the common ratio is outside of this range, the series will be divergent. Additionally, the series will be convergent if the absolute value of the common ratio is less than 1, and divergent if it is greater than or equal to 1.
Simple convergent series have many real-world applications, including finance, physics, and engineering. They can be used to model exponential growth and decay, as well as to calculate compound interest and loan payments. They also play a role in calculating the resistance of electrical circuits and analyzing population growth.