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Tag Archives: analysis
Basic Analysis: Sequence Convergence (3)
So far, we’ve been considering the case where a sequence converges to a real number L. It’s also possible for a sequence to approach +∞ or ∞. The infinity symbol “∞” should be thought of as a convenient symbol instead of … Continue reading
Posted in Notes
Tagged analysis, convergence, limit inferior, limit superior, limits, monotone convergence theorem, sequences, squeeze theorem
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Basic Analysis: Sequence Convergence (2)
Monotone Convergence We start with a useful theorem. Monotone Convergence Theorem (MCT). A sequence is monotonically increasing (or just increasing) if for all n. Now the theorem says: an increasing sequence with an upper bound is convergent. Proof. Let L = sup{a1, a2, … }, … Continue reading
Basic Analysis: Sequence Convergence (1)
Much of analysis deals with the study of R, the set of real numbers. It provides a rigourous foundation of concepts which we usually take for granted, e.g. continuity, differentiation, sequence convergence etc. One should have a mental picture of the … Continue reading
Posted in Notes
Tagged analysis, completeness of reals, convergence, infimum, sequences, supremum
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Number Theory and Calculus/Analysis
Background required: modular arithmetic, calculus. Once in a while, I’ll post something which offers a glimpse into more advanced mathematics. Here’s one. Example 1 For starters, we know from basic algebra that . Let’s see if there’s a corresponding result … Continue reading