Proving Convergence of Sequence an: limn→∞an

Thread starter e.gedge
Start date Jan 15, 2009
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Convergence Sequence

In summary, a sequence in mathematics is a list of numbers that follow a specific pattern or rule. It can be finite or infinite and each number in the sequence is called a term. Convergence of a sequence refers to the behavior of its terms as the sequence progresses to infinity. To prove the convergence of a sequence, one must show that the terms become arbitrarily close to a fixed value as the sequence progresses. This is typically done through mathematical proofs using limit laws and theorems. Proving the convergence of a sequence is important for making predictions and solving problems in various fields. Common methods used to prove convergence include the squeeze theorem, ratio test, root test, and other techniques such as direct comparison, limit comparison, and integral test.

Jan 15, 2009

e.gedge

Let a_n= ( 2n ) 4^-n, for all n greater than or equal to 1
( n )

Prove that sequence a_n converges to a limit, and find lim_n->infinitya_n.

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Jan 15, 2009

e.gedge

that was supposed to be

( 2n )
( n )

Jan 15, 2009

e.gedge

As in, 2n choose n

Jan 16, 2009

HallsofIvy

Science Advisor

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[itex]_{2n}C_n[/itex] is a polynomial in n and 4^-n, being exponential, goes to 0 faster than any polynomial goes to infinity.

What is a sequence in mathematics?

A sequence is a list of numbers that follow a specific pattern or rule. It can be finite or infinite and each number in the sequence is called a term.

What is convergence of a sequence?

Convergence of a sequence refers to the behavior of its terms as the sequence progresses to infinity. If the terms of a sequence approach a single fixed value as the sequence progresses, then the sequence is said to converge.

How is convergence of a sequence proved?

To prove the convergence of a sequence, one must show that the terms of the sequence become arbitrarily close to the limit value as the sequence progresses. This is typically done through mathematical proofs and the use of limit laws and theorems.

What is the importance of proving convergence of a sequence?

Proving the convergence of a sequence is important because it allows mathematicians to determine the behavior of the sequence as it progresses to infinity. This information can be used to make predictions and solve problems in various fields such as physics, engineering, and finance.

What are some common methods used to prove convergence of a sequence?

Some common methods used to prove convergence of a sequence include the squeeze theorem, the ratio test, and the root test. Other techniques such as direct comparison, limit comparison, and the integral test can also be used depending on the specific properties of the sequence.