A converging sequence is a sequence of numbers that approaches a specific value as the number of terms increases. In other words, as more terms are added to the sequence, the numbers get closer and closer to a fixed number, known as the limit.
To prove that a sequence is converging, you must show that the terms of the sequence get closer and closer to the limit as the number of terms increases. This can be done by using mathematical techniques such as the epsilon-delta definition or the squeeze theorem.
{an} is a sequence where each term is multiplied by a constant value, while {an + bn} is a sequence where each term is added to another sequence, {bn}. This means that {an + bn} is a combination of two sequences, while {an} is a single sequence.
The value of {bn} can either help or hinder the convergence of {an + bn}, depending on its behavior. If {bn} is a converging sequence with a limit of 0, then {an + bn} will also converge to the same limit. However, if {bn} is a diverging sequence, then {an + bn} may also diverge or oscillate.
No, a sequence cannot be both converging and diverging at the same time. A sequence can either converge to a specific limit or diverge to positive or negative infinity. However, it is possible for a sequence to appear to be converging but actually be oscillating between two or more values, in which case it is not truly converging.