2005 Student Research Conference:

18th
Annual Student Research Conference

### Mathematics and Computer Science

**Rearrangements of Almost Convergent Sequences**

Christopher C. Bay

Dr. Eric Howard, Faculty Mentor

An infinite sequence of numbers (*x*_{1}, x_{2}, x_{3}, . . .) is said to converge to the number *x* if every interval (*a,b*) around *x*, no matter how small, contains all but a finite number of terms of the sequence. The number *x* is called the limit of the sequence, and the idea of a limit can be generalized in a way that assigns a limit to every bounded sequence. The generalized limit of some sequences can depend on arbitrary choices, but for others it is unique. Such sequences are called *almost convergent*. What happens when the terms of an almost convergent sequence are rearranged? Any rearrangement of a convergent sequence will also converge, but the same is not true for almost convergent sequences. The question of when almost convergence is preserved by a rearrangement is a very interesting one. This question will be addressed, and many examples will be given to illustrate the character of these unusual sequences.

**Keywords:** sequences, limit, almost convergence, rearrangements, infinite series

**Topic(s):**Mathematics

**Presentation Type:** Oral Paper

**Session:** 8-3

**Location:** VH 1408

**Time:** 8:45

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