-
Recent Posts
Archives
- March 2020
- June 2018
- July 2016
- June 2016
- May 2016
- March 2015
- February 2015
- January 2015
- December 2014
- December 2013
- November 2013
- July 2013
- June 2013
- May 2013
- March 2013
- February 2013
- January 2013
- December 2012
- November 2012
- October 2012
- September 2012
- August 2012
- April 2012
- March 2012
- February 2012
- January 2012
- December 2011
- November 2011
- October 2011
Categories
Meta
Pages
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 real-valued 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
Leave a comment
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
Leave a comment
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
Leave a comment
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
Leave a comment
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
Leave a comment
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
