#MathsInMinutes Day 40: Convergent Sequences

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The terms in an ordered list of numbers are convergent if they progressively close in a specific value or limit. But, while we may observe that a sequence seems to be converging on a limit, how can we know what that limit is? For example, methods of estimating pi often rely on a sequence approach. As the sequence gets closer and closer to a number, it would be nice to say that this is the true value of pi.

If a number L is known, then a sequence tends to L if, given any level of error e, there is some stage of the sequence after which all the remaining terms are within e of L. Karl Weierstrass and others discovered that it was not necessary to know L in order to determine whether a sequence converges.

A Cauchy sequence is one in which, given any level of error e, there is some stage in the sequence after which any two points remaining in the sequence are within e of each other. For real numbers, this is equivalent to having a limit.

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