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Real Analysis

Taylor Polynomial

Use Taylor polynomials about 0 to evaluate sin(0.3) to 4dp,showing all workings. 1)F(x)=square root 4+x and G(x)=square root 1+x by writing square root of 4+x=2 square root 1+1/4x and using substitution in one of the standard Taylor series, find the Taylor series about 0 for f.Given explicitly all terms up to term in x raise

Sequences and limit boundaries

1) If (bn) is a bounded sequence and lim(an) = 0, show that the lim(anbn) = 0. Explain why Theorem 3.2.3 cannot be used. Note: Here's Theorem 3.2.3 (a) Let X = (xn) and Y = (yn) be sequences of real numbers that converge to x and y respectively, and let c be an element R. Then the sequences X+Y, X-Y, X∙Y, and cX co

Project Evaluation Review Technique and Critical Path Method

Please refer to the attached file for this PERT / CPM problem: I've determined that the critical path for this network is A - E - F and the project completion time is 22 weeks. Here's where I need help: If a deadline of 17 weeks is imposed, what activities should be crashed?

Sum of series

Determine the sum of the integers among the first 1000 positive integers which are not divisible by 4 or are not divisible by 9. (This is not an exclusive or)

Finding a Function that Satisfies Limits

Find a formula for a function f, that satisfies the following conditions: 1. lim(x->+/-infinity)f(x) = 0, 2. lim(x->0)f(x) = -infinity, 3. lim(x->3-)f(x) = infinity, 4. lim(x->3+)f(x) = -infinity, 5. f(2) = 0.

Infinite Series : Convergence and Divergence (8 Problems)

1) a) Prove that N ∑ 1/n(n+1) = 1- (1/N+1) n=1 Hence, or otherwise, determine whether the following infinite series is convergent or divergent: b) Determine whether each of these infinite series are convergent or divergent. Justify your an

Real Analysis : Continuity, Closed and Open Sets and Differentiability

Prove OR disprove the following statements. Explain. (i) There is a nonempty set S in R ( real numbers) such that S contains none of its limit-point(s). (ii) There is a nonempty set S in R such that S is neither open nor closed. (iii) There is a nonempty set S in R such that S is both open and closed. (iv) Let a

Real Analysis : Differentiability and Sequence of Partial Sums

A). Prove that the function f(x) = e^x is differentiable on R, and that (e^x)' = e^x. ( Hint: Use the definition of e^x, and consider the sequence of partial sums.) My thoughts on a: I tried to prove the differentiability by proving continuity on R, since e^x is series, sum of polynomials, and all polynomials are different

Real Analysis : Lebesgue Integral Problem

Let f be a nonnegative measurable function. Show that (integral f = 0) implies f = 0 a.e. See attached document for notations. Please help: This problem is from Royden's Chap 4 text on Lebesgue Integral.

Central limit theorem

Let's say you work in a company where over 2000 people are employed. Using the Central Limit Theorem, where the mean age of all employed is 37 with a standard deviation of 13; If 5 people are randomly selected, find the probability of their age being less than 22.

40 Problems : Sequences, Series, Convergence, Divergence and Limits

1. For each of the sequences whose nth term is given by the formula below (so of course n takes successively the positive integer values 1,2,3...), does it have a limit as n tends to infinity? In each case, briefly explain your answer including justification for the value of the limit (if it exists) a) (1/3)ⁿ b

Taylor & Maclaurin Series

1- Find the Taylor series generated by f at X = a. f (x) = 1/(10-x) a = 3 2- Find the Maclaurin series for the given functions. A) 1/(6+x) B) sin 10x

Average Value of Continuous Functions and Limits

The definition of average value of a continuous function can be extended to an infinite interval by defining the average value of f on the interval [a, ∞) to be Lim as t approaches ∞ 1/(t-a)integrand from a to t f(x)dx 1. Find the average value of {see attachment} on [0, ∞). 2. Find the lim as x goes t

Convergence or Divergence, Taylor Polynomials, Maclaurin Series and Chain Rule

1. Test for convergence or divergence, absolute or conditional. If the series converges and it is possible to find the sum, then do so {see attachment} 2. Find the open interval of convergence and test the endpoints for absolute and conditional convergence: {see attachment} 3. For the equation f (x) = ... {see attachment

Evaluating a Limit

Thank you in advance for your help; this one sounds like it should be simple, but I still continually get the wrong answer: "Evaluate the limit as x goes to infinity of (1+(3/x))^(4x)."

Real Analysis : Proofs - Uniform Continuity

S(1): Let &#949;=1, and let any &#948;>0 be given. S(2): Let n be an integer > max(1, 1/&#948;), and set x=1/n and y=1/(n+1). S(3): Both x and y belong to (0,1), and |x-y| = 1/n(n+1) < 1/n < &#948;. S(4): However, |f(x)-f(y)| = |n-(n+1)| = 1 = &#949; S(5): This contradicts the definition of uniform continuity (i.e.,