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Real Analysis

Vectors and Limits

See attached file for full problem description. 1) Find the canonic equation of the straight (recta) in the space( espacio), with R 2) Calculate the scaling product of the vectors r y s knowing that r = 3i + 4j; that the module of the vector / s/ = 6; and the angle that both vectors form is 120° 3)Find

General Vector Taylor Series Expansion: Measure of deviation

2. Arfken, p. 342, 5th Ed. (p. 359, 6th Ed.), ) Prob. 5.6.7. Use the General Vector Taylor Series Expansion For a General Function, cI) (r) = (I) (x, y, z) , Of a Three-Dimensional Vector Coordinate, Expressed In Cartesian Coordinates, which is Expanded About the Origin, r = 0 Or x = y = z = 0 , Where 0(0 = (1)(x', y', z') 1 ir,

Stone-Weierstrass Approximation Theorem

Show that there is a countable set F of functions of form x--->a_0 + a_1cosx + a_2cos(2x) + ... +a_ncos(nx) such that every continuous real-valued function on [0,pi] is the uniform limit of a sequence of functions (f_n) in F.

3-Sigma Control Limits

Jim Outfitters makes custom fancy shirts for cowboys. The shirts could be flawed in various ways, including flaws in the weave or color of the fabric, loose buttons or decorations, wrong dimensions, and uneven stitches. Jim randomly examined 10 shirts, with the following results: Shirt Defects 1 8 2 0 3 7 4 12 5 5 6 10

Limits : Proving the Root Test Assuming the Ratio Test

A) Prove root test " lim(sqrt|An)|)=L as n goes to infinity" assuming ratio test "lim(|An+1)|/|A n|)=L as n goes to infinity" ps. {An} is a sequence of non-zero complex numbers b) Prove that although the following power series have R=1 sum(nz^n) does not converge on any point of the unit circle.

Limit

(See attached file for full problem description) evaluate each limit Evaluate:

Finding Area Using Sums and Limits

Given f(x) = x^2 + 3, find the exact area A of the region under f(x) on the interval [1, 3] by first computing n Σf(xi)Δx and then taking the limit as n-->∞. i=1 Please see the attached file for the fully formatted problems.

Problem set For the equation ?(x)= x^(1/2) a) Find the Taylor polynomial of degree 4 of at c = 4 b) Determine the accuracy of the polynomial at x = 2. Question (2) Find the Maclaurin series in closed form of Question (3) Use the chain rule to find dw / dt, where w = x^2 + y^2 + z^2, x=(e^t) cos t, y=(e^t) sin t, z=(e^t), t=0 Question (4): Find the critical points and test for relative extrema: ?(x,y)=2(x^2)+2xy+(y^2)+2x-3

1. For the equation ?(x)= x^(1/2) a) Find the Taylor polynomial of degree 4 of at c = 4 b) Determine the accuracy of the polynomial at x = 2. 2. Find the Maclaurin series in closed form of a) ?(x)=((1) / ((x+1)^2) b) ?(x)=ln ((x^2)+1) 3. Use the chain rule to find dw / dt, where w = x^2 + y^2 + z^2, x=(e^t) cos t, y=(

Series

1.) one of the following series converges absolutely one converges conditionally, the other diverges which is which show steps a.)sum (-1)^n (1/(sqrt(n))) b.) sum (-1)^n (sqrt(n)/(1+sgrt(n))) c.) sum (-1)^n (1/( n(1+sqrt(n))) 2.) test for convergence the sum of ((2n+n^3)/(3+n^4))

Series

1.) list the first 4 terms of the series from n=1 to infinity for (1*3*5***(2n-1))/ (n^2 * n!) 2.) determine the interval of convergence of the power series sum of ((2^n)/(1+n)) * x^n

Geometric 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)

Two sided limit

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

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&#8804;t, s&#8804;b and there exists a constant M such that (a&#8804;s&#8804;b). Then the integral equation has a unique solution when . Please see the attached file for the fully formatted problems.