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Real Analysis

Convergence of Power Series

1 Determine whether the series converges absolutely, converges conditionally, or diverges. ∞ Σ 2∙4∙6∙∙(2n)/2ⁿ(n+2)! n=1 2 Calculate sin 87° accurate to five decimal places using Taylor's formula for an appropriate functio

Power Series Proof

Define the set R[[X]] of formal power series in the indeterminate X with coefficients from R to be all formal infinite sums sum(a_nX^n)=a_0 +a_1X+a_2X^2+... Define addition and multiplication of power series in the same way as for power series with real or complex coeficients,i.e extend polynomial addition and multiplication t

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,

Taylor Series Expansion and derivation of the Euler Formula

Show that (a) sin x = summation (0-infinity) (-1)^n x^(2n+1)/(2n+1)! (b) cos x = summation (0-infinity) (-1)^n x^(2n)/(2n)! Use the Taylor series expansion around the origin, f(x) = summation (0-infinity)[x^n/n!]f^n(0), and derive the power series expansions for sin x , cos x and e^x. Then write out the first few real

Semi-Continuous Function on a Compact Subset

Show that if f is an upper semi-continuous function on a compact subset K of R^p with values in R, then f is bounded above and attains its supremum on K. Edit: R^p--Let p be a natural number and let R^p denote the collection of all ordered "p-tuples"---i.e. (x_1, x_2,..., x_p) with x_i being a real number

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

Proof: Sequences and Limits Example Problem

If {sn}∞ n=1 is a sequence of real numbers such that sn ≤ M for all n and lim n--> ∞ sn =L; prove that L ≤ M. Is the statement true if we replace both inequalities with "<"? See attachment for full equation.

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 &#931;f(xi)&#916;x and then taking the limit as n-->&#8734;. 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=(

Real analysis: Lebesgue Integral

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

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 - open and closed sets

(See attached file for full problem description) 1. In the metric space show that: a. Any open interval of the form (a,b), (a, ), or (- ,b) is an open set. b. A close interval [a,b] is a closed set. c. Any interval of the form [a, ) is a closed set.

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

Limit Proofs

Prove the following a) If lim n-->infinity (a_n*b_n) exists and lim n--> infinity (a_n) exists, then lim n -->infinity (b_n) exists. b) If lim n--> infinity (a_n) = 0 and {b_n} is bounded, then lim n-->infinity (a_n*b_n) exists and equals 0. c) If lim superior (a_n) exists, then {a_n}_n is bounded above.

Sequences and Limits

Consider the real sequence {x_n}_n generated by the iteration scheme x_n+1 = x_n(2-ax_n), for n = 0, 1, 2, ...... where a>0 and x_0 satisfying 0 < x_0 </= 1/a. a. Prove 1/a>/=x_n>0 for all n. b. Prove x_n>/=x_n-1. c. Conclude that lim n-->infinity

Dense Subset, Continuity and Uniform Convergence

Let E C R1 and let D be a dense subset of E. If are continuous real-valued functions on E for n=1,2,..., and fn converges uniformly on D, prove that fn converges uniformly on E. (See attached file for full problem description) I am using the book Methods of Real analysis by Richard Goldberg.

Prove that the Series of Functions Converges Uniformly

(See attached file for full problem description with equations) --- 9.3-5 Let {f_n} (from n - 1 to infinity) be a sequence of functions on [a,b] such that (f_n)'(x) exists for every x is an element of {a,b](n is an element of I) and (1) {(f_n)(x_0)} (from n=1 to infinity) converges for some x_0 is an element of [a,b]. (2

Sequences and Uniform Convergence

Let {fn} infinity-->n-1 be a sequence of continuous real-valued functions that converges uniformly on the closed bounded interval [a, b]. For each n&#1028; I let Fn(x) = &#8747; x--> a fn(t)dt a<x<b Show that {fn} infinity-->n-1 converges uniformly on [a,b]. (Hint: Use 9.2F) Theorem 9.2F;

Mapping, Contraction and Fixed-Point Theorem

(See attached file for full problem description with proper equations) --- 3. Let T(x) = x^2 Show that T is a contraction on (0, 1/3] , but that T has no fixed point on this interval. Does this conflict Theorem 6.4? Explain. Note: We are using the book Methods of Real Analysis by Richard R. Goldberg. This

Fibonacci Sequences, Convergence and Limits

(a) find the first 12 terms of the Fibonacci sequence Fn defined by the Fibonacci relationship Fn=Fn-1+Fn-2 where F1=1, F2=1. (b) Show that the ratio of successive F's appears to converge to a number satisfying r2=r+1. (c) Let r satisfy r2=r+1. Show that the sequence sn=Arn, where A is any constant, satisfies the Fi

Increasing, Bounded, Sequences and Series, Limits and Convergence

The sequence Sn = ((1+ (1/n))^n converges, and its limit can be used to define e. a) For a fixed integer n>0, let f(x) = (n+1)xn - nxn+1 . For x >1, show f is decreasing and that f(x) . Hence, for x >1; Xn(n+1-nx) < 1 b) Substitute the following x-value into the inequality from part (a)

Jacobians of the Algebraic Functions

Independence and relations Real Analysis Jacobians (XI) If u = (y^2)/(2x) and v = (x^2 + y^2)/(2x), find the Jacobian of u,v with respect to x,y. The fully formatted problem is in the attached file.

Real Analysis - Limits of Sequences

Suppose that {an} and {bn} are sequences of positive terms, and that the limit as n goes to infinity of (an/bn) = L > 0. Prove that limit as n goes to infinity of an is positive infinity if and only if the limit as n goes to infinitiy of bn is positive infinity. Here is what I have for proving the first way: Suppose that

Real Analysis: Elementary Sets and Closure

1) Let M be an elementary set. Prove that | closure(M)M | = 0. (closure of M can also be written as M bar, and it is the union of M and limit points of M). 2) If M and N are elementary sets then show that | M union N | + | M intersection N| = |M| + |N| The definition of elementary set : If M is a union of finite members