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Real Analysis

Maclaurin series and Taylor

1.) find maclaurin series for f(x) do not show rn(x)->0 also find radius convergence of f(x)= ln(1+x) 2.) find the taylor series for f(x) centered at the given value of a .. assume a power series expansion, do not show rn(x)-> 0 of f(x)=sin x , a=pie/2 3.) find the sum of the series sum from n=0 to infinity of (3^n

Power Series : Radius and Interval of Convergence

Find the radius of convergence and interval of convergence of the series of the following two problem 1.) sum n=1 to infinity of (n^2*x^n)/(2.4.6.......(2n)) 2.) sum n=1 to infinity of ((2.4.6........(2n))/(1.3.5......(2n-1)))x^n keywords: radii, intervals

Real Analysis Problem

Show that a countable set in R^n is of measure zero. Notes for this section are attached. keywords: measurable, measurability

Real Analysis: Show an integral equation has a unique solution.

Assume that g(t) is continuous on [a,b], K(t,s) is continuous on the rectangle a≤t, s≤b and there exists a constant M such that (a≤s≤b). Then the integral equation has a unique solution when . Please see the attached file for the fully formatted problems.

Testing Series for Convergence

Test for convergence or divergence 1.) sum from n=1 to infinity of (e^1/n)/(n^2) 2.) sum from j=1 to infinity of (-1)^j * ((sqrt j)/(j+5)) 3.) sum from n=2 to infinity of (1/((1+n)^(ln n)) keywords: tests

Real Analysis - Banach Fixed Point Theorem

Prove the following generalization of the Banach Fixed Point Theorem: If T is a transformation of a complete metric space X into itself such that the nth iterate, T^n, is a contraction for some positive integer n, then T has a unique fixed-point.

Real Analysis - Newton's Method and showing convergence.

Newton's Method: Consider the equation f(x)=0 where f is a real-valued function of a real variable. Let x_0 be any initial approximation of the solution and let x_(n+1)=x_n - (f(x_n)/f'(x_n)). Show that if there is a positive number "a" such that for all x in [x_0-a, x_0+a] |(f(x)f''(x))/((f'(x))^2)|<=lambda<1 and |(f(x

Real Analysis - Show E is equicontinuous.

Let E be a set of differentiable functions in C[a,b] with uniformly bounded derivatives; i.e., there exists a number M, independent of f in E, such that |f'(x)|<=M for all x in [a,b] and all f in E. Show that E is equicontinuous.

Real Analysis : Bounded Open Balls

Show that a set E in the metric space X is bounded if and only if, for some "a" in X, there exists an open ball B(a;r) such that E is a subset of B(a;r).

Real analysis metric spaces

(See attached file for full problem description) 7. If d is a real-valued function on which for all x, y, and z in X satistifes d(x,y) = 0 if and only if x=y d(x,y)+d(x,z)&#8805;d(y,z) show that d is a metric on X.

Limit and trigonometric equation

(See attached files for full problem description) 1. Find the limit: lim(t-->0) t^2/(1-cost) 2. solve the following trigometric equation tan(2x) = 2sin(x), where 0<=0< 360 degrees

Sequences and Series (20 Problems): Partial Sums, Convergence and Divergence

Please do all problems below step by step showing me everything. Do simply as possible so I can clearly understand without rework. Adult here relearning so show all work, etc. OK, some said cannot read problems, but do not have a scanner with me know, so typed them in below. Sorry for any problems, but this shopuld clear up

Limits and L'Hopital's Rule

Find the indicated limit make sure you have a indeterminate form before you apply L'Hopital rule (1) lim xgo to0 arctan3x/arcsinx (2) lim x go to pi/2 3secx+5/tanx (3)lim x go to 0 2csc^2x/cot^2x evaluate dx/squrtpix a=0 andb=inifinity

Series : Absolute Convergence

Suppose the summation from k=1 to n of a_k is absolutely convergent and {b_n} is bounded. Prove that this implies the summation from k=1 to n of a_k*b_k is absolutely convergent.

Cauchy Sequence and Limit Supremum

Suppose that {a_n}_n is a real Cauchy sequence. Prove that lim superior n--> infinity (a_n) = lim inferior n--infinity (a_n) so as to conclude that lim n--infinity (a_n) exists.