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Real Analysis

Integral test

1.) use integral test p>1 for series sum of n/((1+n^2)^p) to show that it converges using substitution 2.) a.) explain why sum from n=0 to infinity of (-1)^n (n^2/(1+n^3)) converges b.) how many terms of that series should you sum to have an error no more than (1/100)

Limits

(See attached file for full problem description) Evaluate the following limits (explain how you solved the limits)

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