. If [an is a senies of pobitive terms md [an is convergent then cheek the convergenie of the Sean'ifs E an

Q: Q2 Prove that the scries 2+ x2 3+X2 4+x is Unitormly convergem? in any interval?

A: The given series is: 11+x2+12+x2+13+x2+……. Therefore the given series can be written as:…

Q: 1. Use an appropriate way to determine whether the following is convergent or divergent. k 'Ink i. Σ…

A:

Q: Let a1 = 1 and an+1= √ (an+2). Prove that a nis convergent and find limn→∞ an

A: we have to find limit

Q: For which of the sets below, does the property n > 53 for all x, E S ?imply that (x,) is convergent…

A: which of the following is correct ?

Q: The infinite sum Ea(n) A) S = a(n) has a finite limit B) none of these %3D converges if and only if

A: ∑n=1∞a(n) converges

Q: 3. If [an is a senies of poritive term amd Ian is convesgent ithen cheek the convergenie of the…

A: We are provided that ∑an is a series of positive terms and ∑an is convergent.  The objective is to…

Q: Given: a, = In(n + 1) – In(n) for n E N || Determine if the following is convergent or divergent.…

A: Given  an = lnn+1 - lnn for n∈ℕ.

Q: Consider an = V4n2 + 3n – 2n for n 2 1. - (a) Determine if {an}, n=1 is convergent or divergent (b)…

A: Here, an=4n2+3n-2n, n≥1     =4n2+3n-2n4n2+3n+2n4n2+3n+2n     =4n2+3n-4n24n2+3n+2n     =3n4n2+3n+2n

Q: Given: an = In(n+ 1) – In(n) for n E N | Determine if the following is convergent or divergent. (b)…

A:

Q: Prove: if 0 is less than or equal to r which is less than or equal to 1, then (r^n) converges.…

A:

Q: Let 0 for all n EN. If (x,) is convergent, then lim (x,) = 2 4 Select one: O True O False

A: If xn is convergent, let us assume limn→∞xn=x Also, limn→∞xn+1=x Given, xn(1-xn+1)>14 Taking…

Q: The sequence an = n – Vn +1yn+3 converges to the limit

A:

Q: Let d: a x a R. de fined as d (z. y) = -al. wherer= (),y= () E ra, apace of all sequences convergent…

A:

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

A: We know that ∑k=0∞xkk!=ex. So, we will show that ex does not converge uniformly on ℝ.  Let us assume…

Q: Let {fn(x)} : „x ER and > 0. 1+ nx Show that ift>0, the sequence {fn(x)} converges uniformly on

A: Given: To explain the given statement as follows, To show that if t>0, the sequence fnx converges…

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

A:

Q: Label the following statements as true (T) or false (F). lim, 00 (4n² – 5n) = e If lim,-00 an = 0,…

A: (a)

Q: If > an is convergent and ) b, is divergent, then > (am + bn) is divergent. n=1 n=1 Select one: True…

A:

Q: If lim a, is not 0 then 2 a, is convergent n=1

A: By using theorem we can say that

Q: 1 3. En=1 (2n+1)3

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 by induction that 1 2. Prove that (xn) is contractive and thus convergent. Compute lim (xn).

A:

Q: a) Show by example that E( an /b.) may diverge even thagh E an and E bn converge and no b, equals 0.

A: Since solve the first question for you. If youwant any specific question to be solved then please…

Q: 1 d.x x In (ex)

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: Determine if the given is convergent or divergent. (x²+1)² n Zn=1 (n²+1)²

A:

Q: Show that if $a_{n}>0$ and $\Sigma a_{n}$ is convergent, then $\Sigma \ln \left(1+a_{n}\right)$…

A: Here, we are given that an > 0 and series is convergent. We need to prove that: is convergent.…

Q: k+ln k > ak, where aj = and ak+1 ak k+10 %3D k=1

A:

Q: If E Un is absolutely convergent then (un)² is also n=1 n=1 convergent.

A:

Q: In = Vn (Vn +1- vn

A:

Q: (a) Determine whether {an} is convergent.

A:

Q: Suppose an is conditionally convergent. Define what this means, and then use a proof by n=1…

A:

Q: k en(k) 大=Z

A:

Q: If (am) is increasing and en (an) is bounded by M converges to a, {|a1\, |a|}. max

A: Let (an) is increasing sequence and converges to a,

Q: If lim =c>0 , then b, , a, both convergent and both divergent. in ylgn O ihi O

A: Answer

Q: Let p 3. Prove that if fn converges to f in Lp then f converges to f³ in Lp/3.

A:

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

A:

Q: There is no value of r such that Jo (3t)-r 5 dt converges. 14.

A:

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

A: Given statement is

Q: Determine whether the following statement is true or false. If the statement is true, prove it, and,…

A:

Q: Suppose {a,} is monotonic. Prove that {an} converges if and only if {a} converges

A:

Q: (a) if Ek až is absolutely convergent then E ar is absolutely convergent.

A:

Q: If E, 2"an converges and E o(-1)"2"an diverges, then the radius of conver- gence of o anx" is 2.

A: According to the given information, Suppose,

Q: Given the expression an = V4n² + 3n – 2n for n > 1 Check whether this is convergent or divergent: an…

A: The given sequence is a an=Sqrt(4n2+3n) -2n.

Q: Show that: For any f E C(R/Z; C), the If (n)|² is absolutely convergent, and n=-00

A:

Q: Suppose {f,} is a sequence in Banach space B s.t. |[fn - fn=i|l s /2-). Show that the sequence {f,}…

A: This is a problem of Functional Analysis. Banach space.

Q: Given the expression an = V4n² + 3n – 2n for n >1 Check whether this is convergent or divergent:…

A: solution:-an=4n2+3n-2n   for   n≥1Divergent…

Q: Determine whether {an} is convergent

A: First find the limit of the sequence < an >:

Q: (e) $₁ = 5 and Sn+1 = √√4s +1 for n ≥ 1.

A: Given s1 = 5 and sn+1 = 4sn+1 for n≥1

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.