Math 312, Intro to Real Analysis: Homework Solutions
10 8 Assume that (sn) is a nondecreasing sequence of real numbers Let σn be the average of the first n numbers in our given sequence: σn = s1 +··· +sn n We claim that the sequence (σn) is again nondecreasing
Series - UC Davis Mathematics
4 1 Convergence of series 61 Telescoping series of the form X1 n=1 (a n a n+1) are another class of series whose partial sums S n= a 1 a n+1 can be computed explicitly and then used to study their convergence
01 Degrees
10 Sn hasacontinuousfieldofnon-zerotangentvectorsifandonlyifnisodd Proof Suppose x7v(x) is a tangent vector field on Sn, assigning to a vector x2 Sn the vector
Chapter 3
22 3 Continuous Functions If c ∈ A is an accumulation point of A, then continuity of f at c is equivalent to the condition that lim xc f(x) = f(c), meaning that the limit of f as x → c exists and is equal to the value of f at c
The Definition of a Manifold and First Examples
WOMP 2012 Manifolds Jenny Wilson The Definition of a Manifold and First Examples In brief, a (real) n-dimensional manifold is a topological space Mfor which every point x2Mhas a neighbourhood
Homework { corrections
Math 320: Analysis I Spring 2015 By the Completeness Axiom, supfs n: n > Ngand supft n: n > Ngexist and are real numbers Since supfs n: n > Ngand supft n: n > Ngare the least upper bounds for fs n: n > Ngand ft n: n > Ng, respectively, the last inequality in (1) implies s n+ t n supfs n: n > Ng+ supft n: n > Ng: (3)
Trig Cheat Sheet - Pauls Online Math Notes
Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p
Math 412 The Orbit Stabilizer Theorem
(c)Karen E Smith 2018 UM Math Dept licensed under a Creative Commons By-NC-SA 4 0 International License Math 412 The Orbit Stabilizer Theorem Fix an action of a group G on a set X For each point x of X, we have two important concepts: DEFINITION: The orbit of x 2X is the subset of X O(x) := fg xjg 2GgˆX:
Mathematical induction & Recursion
Recursive definition of * • Basis step: – empty string * • Recursive step: –I wf * and x then wx * M Hauskrecht Length of a String Example: Give a recursive definition of l(w), the length of the string w Solution: The length of a string can be recursively defined by: l(‘’) = 0; the length of the empty string
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