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Infinite Geometric Series:
A geometric series is really the sum of an infinite number of terms with a constant ratio between successive terms in mathematics. A geometric series has been generally written as a + ar + ar2 + ar3 + …, where a represents the coefficient of each term and r represents the common ratio of adjacent terms. The geometric series is one of the most basic examples of infinite series and can be used to introduce Taylor and Fourier series. Geometric series played an essential role in the development of calculus, and they are used throughout mathematics. They also have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.
Overall, determining whether a given sequence is geometric is as simple as determining whether successive entries in the sequence all have the same ratio. A geometric series’ common ratio can indeed be negative, resulting in an alternating sequence. An alternating sequence contains numbers that alternate between positive and negative signs. The sort of behaviour of a geometric sequence is determined by the common ratio.
The sum of an infinite geometric sequence yields an infinite geometric series. There would be no final term in this series. The sum of all finite geometric series could be found. When the common ratio in an infinite geometric series is greater than one, the terms in the sequence grow larger and larger and adding the larger numbers yields no final answer. Infinity is the only possible answer. As a result, we do not consider the common ratio greater than one for an infinite geometric series.
We can have the sum of an infinite geometric series if the common ratio r is between 1- and 1. That really is, the sum for | r |<1 exit.
The formula gives the sum S of an infinite geometric series with -1<r<1.
A convergent series is indeed an infinite series with a sum, and the sum Sn is known as the partial sum of the series.
An infinite series could be represented using sigma notation.
FAQs:
Does the infinite geometric series diverge or converge explain?
Geometric series with a finite number of terms are always convergent, whereas infinite geometric series are not always convergent. The magnitude of the common ratio (r) determines the convergence of an infinite geometric series. When it's less than one, the series will converge; otherwise, it will diverge.
Under what conditions do an infinite geometric series converges?
A simple test could be used to determine whether a geometric series converges or diverges; if −1
What is the difference between infinite sequence and infinite series?
An infinite sequence of numbers is indeed a list of numbers that has an infinite number of numbers in it. An infinite series has been defined as the sum of an infinite sequence.
