A limit proof on sequence convergence is a mathematical technique used to determine the behavior of a sequence as its terms approach a certain value, known as the limit. It involves showing that the terms of the sequence get closer and closer to the limit as the index of the terms increases.
To prove that a sequence converges to a specific limit, you need to show that for any small positive number, there exists a corresponding index of terms in the sequence after which all terms are within that small positive number of the limit. This can be done using the formal definition of convergence or through other convergence tests such as the squeeze theorem.
No, a sequence can only converge to one limit. If a sequence has multiple limits, it is considered to be divergent.
A convergent sequence is one that approaches a specific limit as the index of the terms increases, while a divergent sequence does not approach any specific limit. Divergent sequences can either have terms that increase or decrease without bound, or have terms that oscillate between different values.
Some common methods used in limit proofs on sequence convergence include the formal definition of convergence, the squeeze theorem, and the monotone convergence theorem. Other techniques, such as using algebraic manipulations or the ratio test, may also be useful in certain cases.