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Monthly Archives: November 2012
Basic Analysis: Differentiation (1)
In this article, we’ll look at differentiation more rigourously and carefully. Throughout this article, we suppose f is a realvalued function defined on an open interval (b, c) containing a, i.e. f : (b, c) → R with b < a < c. Theorem. The derivative of f(x) at a is … Continue reading
Posted in Notes
Tagged advanced, analysis, analytic, chain rule, differentiation, taylor series
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Basic Analysis: Limits and Continuity (3)
Let’s consider multivariate functions where . To that end, we need the Euclidean distance function on Rn. If x = (x1, x2, …, xn) is in Rn, we define: Note that x = 0 if and only if x is the zero vector 0. Now we are ready … Continue reading
Posted in Notes
Tagged advanced, analysis, continuity, limits, multivariate, open balls, open subsets, topology
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Basic Analysis: Limits and Continuity (2)
Previously, we defined continuous limits and proved some basic properties. Here, we’ll try to port over more results from the case of limits of sequences. Monotone Convergence Theorem. If f(x) is increasing on the open interval (c, a) and has … Continue reading
Posted in Notes
Tagged advanced, analysis, continuity, limits, monotone convergence theorem, points of accumulation, squeeze theorem
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Basic Analysis: Limits and Continuity (1)
[ This is a continuation of the series on Basic Analysis: Sequence Convergence. ] In this article, we’ll describe rigourously what it means to say things like . First, we define a punctured neighbourhood of a real number a to be … Continue reading
Basic Analysis: Sequence Convergence (4)
In this article, we’ll consider the convergence of an infinite sum: . We call this sum an infinite series. Let be the partial sums of the series. Definition. We say that is L (resp. ∞, ∞) if the partial sums converge to … Continue reading
Posted in Notes
Tagged abel transformation, alternating series, analysis, basel series, convergence, series, telescoping series
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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