### Geometric and Power Series

By manipulating the geometric series for the sum of n=0 to infinity for x^n=1/(1-x) for |x| < 1 determine the power series (about 0) for x/(2+3x).

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By manipulating the geometric series for the sum of n=0 to infinity for x^n=1/(1-x) for |x| < 1 determine the power series (about 0) for x/(2+3x).

Find the two-sided limit lim f(x) x -> 2 See the attached file.

(See attached file for full problem description)

(See attached file for full problem description) Find the limit using the limit theorems lim 6x x->3

(See attached file for full problem description) Find the limit using the limit theorems - ___ lim √4x x16

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).

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

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

Prove theorem 7.3 in notes attached. Section 7: The Lebesgue Integral Definition 7.1 Let L be the set of real-valued functions f such that for some g and h in f=g-h almost everywhere. The set L is called the set of Lebesgue integrable function on and the Lebesgue integral of f is defined as follows: . Theorem 7

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

Show that a countable set in R^n is of measure zero. Please see attached for full problem.

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Limits of sequence. See attached file for full problem description.

Let be defined by . Use lower integral and upper integral to determine the Riemann integrability of f on [0,1]. Please see the attached file for the fully formatted problems.

If f is a function from R to R, and there exists a real number aE(0,1) such that |f'(x)|≤a for all xER , show that the equation x = f(x) has a 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.

If E is equicontinuous in C(X,R), show that E-bar (the closure of E) is also equicontinuous. keywords: equicontinuity

Assume that f is a continuous real valued function on the compact space X, then show there exists a point x-bar E X such that f(x-bar)=inf{f(x): x E X). See the attached file.

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

If f is a function from R to R which is increasing on [a,b], show that f is Riemann integrable on [a,b].

Let X be the set of all continuous functions from I_1=[t_0-a_1, t_0+a_1] into the closed ball B[g(t_0);b] is a subset of R_n. Show that for each a>0 the rule d(x,y)=max(|x(t)-y(t)|e^(-a|t-t_0|)) defines a metric on X and that the metric space (X,d) is complete.

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.

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

Show that a contraction is continuous.

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.

Let (X,d) be the metric space consisting of m-tuples of real numbers with metric d(x,y)=max{|a_k-b_k|:k=1...m} where x={a_1, a_2,...,a_m} and y={b_1, b_2,...,b_m}. In this space is every closed and bounded set compact? keywords: Heine-Borel, Borel

Show that if f is a continuous real-valued function on the compact space X, then there exist points x_1, x_2 in X such that f(x_1)=inf{f(x):x in X} and f(x_2)=sup{f(x):x in X}.

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).

Let X be the set of all bounded sequences of real numbers. If x=(a_k) and y=(b_k) let d be the metric funtion defined by d(x,y)=sup{|a_k - b_k|} (note _ denotes subscript) Show that the metric space defined above is complete.

Let (X,d) be a metric space. Define a closed ball with center x and radius r to be the set B[x;r]={y:d(x,y)<=r}. Prove that B[x,r] is a closed set.