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Test the series for convergence or divergence by using the Comparison Test or the Limit Comparison Test.
Prove that every infinite and bounded point collection in the plane (R2) has a limit point.
Show that the nth derivative of f(x) exists for all n ? N. Please justify all steps and be rigorous because it is an analysis problem.
Prove that there exists a constant Ca such that log x = Caxa for all x ? [1,8), Ca ? 8 as a ? 0+, and Ca ? 0 as a ? 8.
Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrability on R, where they define partition.
What is the maximum of F = x1 + x2 + x3 + x4 on the intersection of x21 +x22 +x23 + x24 = 1 and x31+ x32+ x33+ x34= 0?
Write a composition series for the rotation group of the cube and show that it is indeed a composition series.
Find MacLaurin Series for the given function f. Use the linearity of the Laplace Transform to obtain a series representation L(f)=F(s)
Prove that f(x)<=0 for all x in [a,b]. Is this true if we require only integrability of the function?
How can I provide a geometric interpretation of this formula in terms of areas and then prove this formula.
Linear Programming Model for Maximizing Profit of a Production Schedule
Suppose that the function f:[0,*)->R is continuous and strictly increasing, with f(0) = 0 and f([0,*)) = [0,*).
Maximizing the Revenue for an Airline Company.An airline has a new airplane that will be fitted out for a combination of first and second class passengers.
Let f:[a,b] mapped to the Reals be a function that is integrable over [a,b], and let g:[a,b] mapped to the Reals be a function that agrees.
Use mean value theorem to prove that (inf U(f,g,p), for p is element of P) = (sup L(f,g,p), for p is element of P) = ( INTEGRAL f(x)g'(x)dx, as x from a to b)
Infer that the nth root of a natural number is either a natural number or it is irrational.
Find an article through newspapers, magazines, professional journals, etc and find at least two examples of data that are best modeled using linear formulae.
Show that the sum from 0 to infinity of (1-x)x^n does not converge uniformly on [0,1]. What subintervals of [0,1] does it converge uniforlmly on?
Given a function f and a subset A of its domain, let f(A) represent the range of f over the set A; f(A)={f(x) : x belong to A}.
Show that if sum x_n converges absolutely and the sequence(y_n) is bounded then the sum x_n y_n converges.
Assume that A And B are nonempty, bounded above and satisfy B subset or equal of A. Show that sup B<= sup A
If A1,A2,A3,...,Am are each countable sets, then the union A1 U A2 U A3...U Am is countable.
Assume (a_n) is a bounded sequence with the property that every convergent subsequence of (a_n) converges to the same limit a belong to R.
Assume a_n and b_n are Cauchy sequences.Use a triangle inequality argument to prove c_n=Absolute value of a_n-b_n is Cauchy.
Show that if sum a_n converges absolutely then sum a^2_n also converges absolutely.Does this proposition hold without absolute converge.