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ctg81
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Could anyone please explain "convergent sequence" with example.
A convergent sequence is a sequence of numbers that approaches a certain value as the sequence goes on. This value is called the limit of the sequence. For example, the sequence 1, 1/2, 1/4, 1/8, ... approaches 0 as the sequence continues.
To prove that a sequence is convergent, you must show that as the sequence goes on, the numbers get closer and closer to a certain value. This can be done by using techniques such as the squeeze theorem or the limit definition of convergence.
One example of a convergent sequence is the Fibonacci sequence, where each number is the sum of the two previous numbers (1, 1, 2, 3, 5, 8, ...). As the sequence goes on, the ratio between consecutive terms approaches the golden ratio, approximately 1.618.
A convergent sequence approaches a certain value as it goes on, while a divergent sequence does not. In other words, a convergent sequence has a limit, while a divergent sequence does not have a limit. An example of a divergent sequence is the sequence 1, 2, 3, 4, ..., which goes on infinitely without approaching a specific value.
Convergent sequences are used in many areas of science, such as physics, engineering, and economics. For example, in physics, convergent sequences can be used to model the motion of a falling object or the growth of a population. In economics, they can be used to model the convergence of prices or interest rates over time.