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In mathematics, a sequence is an ordered list of numbers. A converging sequence is a sequence of numbers that approaches a finite limit as the number of terms in the sequence increases. This means that as you add more terms to the sequence, the values in the sequence get closer and closer to a specific number.

For example, consider the sequence {1, 1/2, 1/4, 1/8, …}. As the number of terms in the sequence increases, the values get closer and closer to 0. This sequence converges to 0, meaning that 0 is the limit of the sequence.

A converging sequence can be formally defined as follows:

Let {a_n} be a sequence of real numbers. We say that {a_n} converges to a real number L if, for every positive number ?, there exists a positive integer N such that |a_n – L| < ? for all n ? N.

In other words, no matter how small a positive number ? you choose, there is a point in the sequence beyond which all the terms are within ? of the limit L. This is also known as the epsilon-delta definition of convergence.

Intuitively, a converging sequence is one where the values in the sequence eventually settle down and get arbitrarily close to a fixed value. This fixed value is known as the limit of the sequence. A sequence that does not converge is said to diverge.

The limit of a converging sequence is unique, meaning that if a sequence converges, it can only converge to one specific value. This is because if a sequence converges to two different values, then the difference between those two values must be greater than 0. But if the sequence is converging, then there must be a point beyond which all the terms are within ? of both values, which is a contradiction.

There are many different techniques for proving that a sequence converges. Some of the most common techniques include the squeeze theorem, the ratio test, and the root test.

The squeeze theorem is a powerful technique for proving convergence that relies on finding two other sequences that “squeeze” the original sequence from above and below. If the two squeeze sequences converge to the same limit, then the original sequence must also converge to that limit.

The ratio test and the root test are both techniques for determining whether a series converges or diverges. A series is simply the sum of the terms in a sequence. If the terms in a sequence are positive, then the series is also positive.

The ratio test involves taking the limit of the ratio of consecutive terms in the sequence. If this limit is less than 1, then the series converges. If it is greater than 1, then the series diverges. If it is exactly 1, then the test is inconclusive.

The root test involves taking the limit of the nth root of the nth term in the sequence. If this limit is less than 1, then the series converges. If it is greater than 1, then the series diverges. If it is exactly 1, then the test is inconclusive.

In addition to these techniques, there are many other tools that mathematicians use to prove convergence, including the Cauchy criterion, the monotone convergence theorem, and the Bolzano-Weierstrass theorem.

One important property of converging sequences is that they are bounded. This means that there exists a number M such that |a_n| ? M for all n. This follows from the definition of convergence, since if a sequence is unbounded, then it cannot approach a finite limit.

Definition:

A sequence {an} is said to converge to a limit ‘L’ if and only if for every positive number ? (epsilon), there exists a positive integer N such that for all n greater than N, the absolute difference between an and L is less than ?. Mathematically, it can be represented as,

lim (n ? ?) an = L

The above equation means that as the number of terms in the sequence approaches infinity, the value of an gets closer and closer to the limit L.

Examples:

• Consider the sequence {1, 1/2, 1/3, 1/4, …}. It can be observed that the terms of the sequence are getting smaller and smaller. As the number of terms in the sequence approaches infinity, the value of the terms approaches zero. Thus, the sequence converges to zero.
• Consider the sequence {1, 2, 4, 8, 16, …}. It can be observed that each term of the sequence is double the previous term. As the number of terms in the sequence approaches infinity, the value of the terms grows without bound. Thus, the sequence diverges.
• Consider the sequence {(-1)^n}. The terms of the sequence alternate between -1 and 1. As the number of terms in the sequence approaches infinity, the sequence does not approach a single value. Thus, the sequence diverges.
• Consider the sequence {1/n^2}. It can be observed that the terms of the sequence are getting smaller and smaller. As the number of terms in the sequence approaches infinity, the value of the terms approaches zero. Thus, the sequence converges to zero.
• Consider the sequence {cos(n?/4)}. The terms of the sequence oscillate between -1 and 1. As the number of terms in the sequence approaches infinity, the sequence does not approach a single value. Thus, the sequence diverges.

Properties of Converging Sequences:

• Uniqueness of Limit: A sequence can have only one limit. If a sequence has more than one limit, then the sequence is said to be divergent.
• Limit is a Constant: The limit of a convergent sequence is a constant. This means that the value of the limit does not depend on the first few terms of the sequence.
• Algebraic Operations: If two sequences {an} and {bn} converge to limits L and M respectively, then the sequences {an + bn}, {an – bn}, and {an x bn} also converge to limits L + M, L – M, and L x M respectively.
• Squeezing Theorem: If {an}, {bn}, and {cn} are sequences such that an ? bn ? cn for all n greater than some fixed integer N, and {an} and {cn} converge to the same limit L, then {bn} also converges to L.
• Monotonic Sequences: A sequence is said to be monotonic if its terms either increase or decrease. If a monotonic sequence is bounded, then it converges to a limit.