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2 Sequences: Convergence and Divergence

2 Sequences: Convergence and DivergenceIn , we consider (infinite) sequences, limits of sequences, andbounded and monotonic sequences of real numbers. In addition to certainbasic properties of convergent sequences, we also study divergent sequencesand in particular, sequences that tend to positive or negative infinity. Wepresent a number of methods to discuss convergent sequences together withtechniques for calculating their limits. Also, we prove thebounded monotoneconvergence theorem(BMCT), which asserts that every bounded monotonesequence is convergent. In , we define the limit superior and thelimit inferior. We continue the discussion with Cauchy sequences and give ex-amples of sequences of rational numbers converging to irrational numbers. Asapplications, a number of examples and exercises are Sequences and Their LimitsAn infinite(real) sequence (more briefly, a sequence ) is a nonterminatingcollection of (real) numbers consisting of a first number, a second number,a third number, and so on:a1,a2,a3.

Sep 23, 2016 · Sequences: Convergence and Divergence In Section 2.1, we consider (infinite) sequences, limits of sequences, and bounded and monotonic sequences of real numbers. In addition to certain basic properties of convergent sequences, we also study divergent sequences and in particular, sequences that tend to positive or negative infinity. We

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