Which of the following is a condition for the convergence using the Fixed-Point Iteration Method? O F(x) > 0 O F(x) > 1 O F(x) < 1 O F(x) < 0
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- Suppose we have the sequence of functions $f_n(x)=x^n$ defined on $[0,1],$ and suppose $f_n\to f$ pointwise where $f(x)=0$ if $x\in(0,1]$ and $f(x)=1$ if $x=1.$ Prove that Uniform Convergence fails.Prove the following statement: If (fn) and (gn) are uniformly convergent sequences of functions, then (fn + gn) converges uniformly.Note that the following three Fixed-Point Iterations converge to √2. A) x → (1/2)x + 1/x B) x → (2/3)x + 2/(3x) C) x → (3/4)x + 1/(2x) Which of the following rank correctly the ones that converge from fastest to slowest? Group of answer choices a.) B)→A)→C)B)→A)→C) b.) For all, the convergence speed are same. c.) A)→B)→C)A)→B)→C) d.) C)→A)→B)C)→A)→B) e.) C)→B)→A)
- Find the first few coefficients (c0, c1, c2, c3, c4,) and radius of convergence (R).(c) Carefully construct a nested family of subsequences (fm,k), and show howthis can be used to produce a single subsequence of (fn) that convergesat every point of A.Find the interval of convergence for the following
- Give an example of a sequence of differentiable functions {f_n: (-1,1)—>R} that converges uniformly but for which {f’_n(0)} is unbounded.Show that the sequence (x)=(1/√n) convergence.a) Suppose (an) is Cauchy and that for every k ∈ N, the interval (−1/k, 1/k) contains at least one term of (an). Can we say that (an) converges to 0? Either show that it does or give a counter-example.
- Prove that a bounded decreasing sequence {xn} converges by using the e− N definition ofconvergence and appealing to the fact that a bounded increasing sequence must converge.Let fn(x) = x/(n^2+x^2) for x ∈ R. Show that the sequence {fn} converges uniformly to the function that is everywhere zero.. Let gn = nχ[1/n,2/n] and g = 0. Show thatZ 20g 6= limn→∞ Z 20gn.Does the sequence (gn) converge uniformly to g? Does the Monotone Convergence Theoremapply? Does Fatou’s Lemma apply?