A converging series is a sequence of numbers that approaches a specific limit as more terms are added. In other words, the sum of the terms in the series will get closer and closer to a specific value as the number of terms increases.
To determine if a series is converging or diverging, you can use a variety of tests such as the Ratio Test, the Root Test, or the Comparison Test. These tests compare the given series to a known convergent or divergent series to determine its behavior.
The formula for the sum of an infinite series is given by S = a/(1-r), where S is the sum, a is the first term, and r is the common ratio between consecutive terms. However, this formula only works for geometric series with a common ratio between -1 and 1.
To solve this sum, we can use the Ratio Test. We can rewrite the series as (4/3)^n + (5/3)^n and then compare it to the convergent geometric series (5/3)^n. Since the ratio between consecutive terms in our series is always less than 1, the series converges.
One example of a converging series is the geometric series 1/2 + 1/4 + 1/8 + 1/16 + ... This series approaches a limit of 1 as the number of terms increases.