(a) What is a sequence? (b) What does it mean to say that limn →∞ an=8 ? (c) What does it mean t

step 1 okay. Sequence is an ordered collection of numbers where we're allowed tohave repeats of the sam

(a) What is a sequence? (b) What does it mean to say that...

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Key Concepts

Sequence

A sequence is an ordered list of elements, typically numbers, where each element is associated with a unique natural number index. It can be defined as a function from the natural numbers to a set, often the set of real numbers, allowing one to analyze its behavior as the index increases.

Limit of a Sequence Converging to a Finite Value

When we say that the limit of a sequence is a finite number, such as 8, we mean that for every positive distance (?) no matter how small, there is an index N such that for every index n greater than N, the distance between the sequence term a? and 8 is less than ?. This formalizes the idea that the terms of the sequence get arbitrarily close to the value 8 as n increases.

Limit of a Sequence Diverging to Infinity

Declaring that a sequence's limit is infinity means that the terms of the sequence grow without bound. More precisely, for every real number M, no matter how large, there exists an index N such that for every index n greater than N, the sequence term a? is greater than M. This captures the idea that the sequence eventually exceeds any finite value.

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