7. Suppose A and B are compact subsets of R and that fn is a sequence of functions converging uniformly on both A and B separately. Show that fn converges uniformly on AU B.

Q: |an also converges because 0 If an converges then < lan|, for n = 1, 2, ... n=1 n=1

A:

Q: 3. Suppose {fa) is an equicontinuous sequence of functions on a compact set K, and U.) converges…

A:

Q: If 2a, converges, then 1. diverges Select one: True False

A: Use the defination of convergence of a series.

Q: Consider the sequence n²x(1 – nx) x € [0, 1/n] { - fn(x) := elsewhere . Does it converge uniformly…

A:

Q: 4. Consider the sequence of functions : f₁(x) = 1 n³ [x−²+ +1 (a) the uniform topology (If yes, to…

A: The given sequence of function is ;    fn(x) = 1n3x-1n2+1 We have to answer the following questions…

Q: Prove that the sequence given by 2.4. 6.(2n) Un = 3.5.7.…(2n + 1) converges and find its limit.

A: Given query is to prove that sequence converges.

Q: Show that the sequence f.(x) = xn converges for each xE[0,1] but that the convergence is not…

A:

Q: 3.3 Prove that does not converge uniformly on R. k! k=0

A:

Q: 4. Let f: DCR→ R be uniformly continuous and {xn} be a Cauchy sequence in D. Then show that {f(Xn)}…

A:

Q: 5. Use the Limit Comparison Test to show that E=1 1+k2 1 100 Lk%31 converges.

A:

Q: 2. Show that if a > 0, then the sequence (n²a?e¬n*) coverges uniformly on the interval [a, ∞), but…

A:

Q: an Show that if lim =0 and E-i b, converges then En=1 converges too. an =1 n+ 00

A: Given that  limn→∞anbn=0  and  ∑n=1∞bn converges.

Q: Consider the sequence n²r(1 – nx) x€ (0, 1/n] =: (x)"f elsewhere . Show it converges uniformly on…

A: Consider the sequence  fnx=n2x1-nx      ,x∈0,1n0                       ,elsewhere The limit function…

Q: If o an is convergente then o nan is divergente Select one: O True O False

A:

Q: What are all of the values of x ,such that - (-1) n+A converges? n=1 .1 X26 .2 06

A:

Q: Is {3n/(3n-1)} n=1 to infinity, a convergent sequence in R with the usual metric dR(x,y) = abs(x-y)?

A:

Q: . Find all values of x for which пх" n=1 (n + 1)(2x + 1)" converges absolutely.

A:

Q: 1 3 .5 4 X +x

A:

Q: If 2 a, converges, then la,| diverges. n=1 n=1 True False

A: Given statement is FALSE. I HAVE given a counter -example.

Q: For each n ≥ 1, let fn : R → R, fn(x) =nx/(n + x^2) Show that the sequence of functions {fn}n≥1…

A:

Q: Show that the sequence {fn} where fu(r) = e-n* is uniformly convergent on [a, b), a > 0.

A:

Q: 13.D. Show that, if we define fn on R by fa (x) 1+ n?x? then (fn) converges on R.

A: givenfn(x)=nx1+n2x2dividing numerator and denominator by x2fn(x)=nxx21+n2x2x2

Q: ) Prove that if XnXnand YnYn are any two sequences of real numbers such that XnXn converges to xx…

A:

Q: 4. Show that the sequence {fn}, where fn(x) = xER 1+n°x*' is point-wise convergent but is not…

A:

Q: a. In a metric Space (X, d) if (xn) and (yn) are two sequences in X such that : X - x€ X and yn → y…

A: a) In a metric space (X,d) Let (xn) and (yn) be two sequences such that xn -&gt; x and yn -&gt; y…

Q: 4. Show that the sequence {fn}, where nx fn(x) = xER 1+n°x is point-wise convergent but is not…

A:

Q: Suppose {f,} is a sequence in Banach space B s.t. ||fn - fn+1|| s'/2-). Show that the sequence {f,}…

A: For a vector space X, over a field F, a normed linear space X,· is referred to as Banach space if…

Q: 4. Let f : D CR → R be uniformly continuous and {xn} be a Cauchy sequence in D. Then show that…

A:

Q: 5. (a) Suppose 1 and Xn+1 = 2 Show that {n} converges and find the limit. X1 for all nEN.

A:

Q: 4. Let f : DCR → R be uniformly continuous and {In} be a Cauchy sequence in D. Then show that {f(1)}…

A: Image of a Cauchy sequence

Q: .8.8 Suppose that Cn >0 for each n E N. Prove that n either converges = diverges to infinity.

A:

Q: Let f : A → R and (xn),(Yn) are two sequences converges to c such that Xn, Yn # c, Vn E N. If…

A: We understand this statment by the help of an counterexample .

Q: Use Cauchy Criterion to prove that the sequence {sn} converges where Sn + Sn-1 Sn+1 n > 1, 2 and…

A:

Q: Show that the sequence ffn}, where fnox) = ne is uniformly convergent nx+1 on [a, b], aso but is…

A: Given: fnx=nxnx+1 To show that the given sequence is uniformly convergent on a,b, a&gt;0

Q: Show by example that E(a,/b,) may converge to something other than A /B even when A = equals 0. Σαν…

A:

Q: Let fn(x) x +n b. Use the Lemma for non-uniform convergence to show that the sequence does not…

A:

Q: Suppose {fn}n=1 converges uniformly on E. Show {fn}n=1 converges pointwise on E.

A:

Q: Is the following statement correct: If (fn) is a sequence of differentiable functions on [a, b] such…

A: Given statement is

Q: Show that the fractional parts of a given sequence {annz1 is uniformly distributed if and only if 1…

A:

Q: uniformly co

A: Given, if fn and gn are uniformly convergent sequences of functions. To prove that fn+gn converges…

Q: 5) Find all values of x for which En=1 nxn (n+1)(2x+1)n converges absolutely.

A:

Q: an Show that if lim bn =0 and =1 n- 00 converges then En=1ªn converges too. n=D1

A: We have to solve given problem:

Q: Show that 2"x" converges if |x| <. n=1

A:

Q: Suppose that (an) and (bn) are convergent sequences of real numbers with limits an → a and bn → b,…

A:

Q: The set S on wc fn Cx) = 1+ x" Converges pointwire ?

A:

Q: 20. (a) Suppose that (u2n) and (u2n+1) converge to the same limit e. Prove that (un) converges to l.

A: Since you have asked multiple questions, we  will solve first question for you. If you want other…

Q: 4. Show that the sequence {fn}, where nx fn(x)=- xER 1+n°x' is point-wise convergent but is not…

A:

Q: Consider the sequence of functions {fn}n=2 where fn: [a, b] → R defined for n > 2 by if 0 sxs fa =…

A:

Q: 4. Show that the sequence (n) does not converge uniformly on [0, 2] by showing that the limit…

A:

Q: O Find the set s on w/c In )= x"(1-x") converges pointwise.

A:

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.