The sequence a=n-vn+1Vn+3 converges to the limit A) -3 B) -1 C) 1 D) +2 E) -2 F) 0

Q: Exercise 10. Let {an}=1 be a sequence converges to a limit L E R. Prove that any subsequence of…

A: Given a sequence ann=1∞ is converges to a limit L∈ℝ  Let <bnk> be a subsequence of…

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

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Q: Show that the sequence {f}, where f₁²x) = tan `nx, x70 is uniformly ノ on any interval [a,b], aso but…

A: Given function as fnx=tan-1nx, x≥0

Q: Let f(r) = > anx" converges on an interval (-R, R) and let rn be a non zero sequence in n=0 (-R, R)…

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Q: Consider the set U = {(x, sin(1/x)) : 0 < x} C R². a) Show that every point on the vertical line…

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Q: Let X be a set, and define d(x,y) ={0, if x=y; 1, otherwise A sequence (xn) ⊆ X is called…

A: Convergent of a sequence: Let (X,d) be a metric space, then a sequence {xn} is said to be convergent…

Q: If 2n=1a, converges, and if a„ # 1 and a,n > 0 for all n, a. Show that E=1a, converges. b. Does…

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Q: Is the sequen ce in (R.To)? (0,1.0.1.0,1,..) convergent Prove the answer

A: It is given that the topology space ℝ,τ0. Here, τ0=ℝ,∅. That is, the open sets of ℝ  are ℝ and ∅.…

Q: Let xn be a sequence in a metric space X. Suppose that x, has a subsequence xn, converges to a point…

A: Given, xn  be a sequence  in a metric space X and  xn has a subsequence xnjthat converges to  a…

Q: Establish the convergence or divergence of the given sequenoes in full detail: (a) In = n!. (b) %. =…

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Q: Let (X,) be a sequence of real numbers. If for some x e R, (x,) converges to both 7x + 2 and 2x - 13…

A: option  (d)  is correct

Q: Using Sandwich Theorem, the sequence 1 cos? n + 1 an = 4 - n converges to 4 converges to 2 diverges…

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Q: The sequence a, converges to the limit A) e B) ve C) Ve D) e2 E) F) e2

A: Given : The sequence     To find the convergence of the given sequence.

Q: The sequence ɑn = 1 + converges to the limit A) e B) ve C) Ve D) e? E) F) e? /2

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Q: DETAILS The sequence an = converges to the limit A) e B) ve C) Ve D) e2 E) 5 F) e2

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Q: Using Sandwich Theorem, the sequence 2 cos? n + 1 an = 4 - diverges converges to 0 converges to 2…

A: Use range of cos x which lies in -1 and 1

Q: Calculate the Radius and the Interval of Convergence na" 2n n=0

A: we will be using ratio test for the series to converge.

Q: Find the radius of the given puer of convergence (2n)! 2n %3D n=I (n!)2

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Q: Consider the sequence {fn(x)}, where sin(nx) fn(x) = 1+ nx a. Show that fn(x) converges pointwise to…

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Q: Suppose {pn} n=1 to infinity, is a Cauchy sequence in a metric space, and some subsequence {pni} i=1…

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Q: Find the radius of convergence for E, (11/ 5)" 2n

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Q: Using Sandwich Theorem, the sequence 2 an = 4 – Cos n2 + 1 - O diverges O converges to 4 converges…

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Q: State the definition of a sequence rn converging to a point r in a metric space (X, d). Prove that…

A: I have given the definition of the conference sequence in metric space and that used for given…

Q: Using Sandwich Theorem, the sequence 2 1. an = 4 sin? 3 %3D n + 1 converges to 0 converges to 2…

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Q: Show that if A is a real number such that A > 0, then the sequence (xk) defined by A 2xk 1 Xk+1 k >0…

A: Given, A∈ℝ, A&gt;0, xk is defined by, xk+1=12xk+A2xk ,k≥0....1x0≥A2  and xk≥0 ∀k≥0 Now,…

Q: 3- a- By definition show that (x,) = (+3) converges to 3. b- Does (+3) bounded, Cauchy? Show by part…

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Q: Determine whether b, converges or diverges. n=1 п +1 a. bị = 1, bat1 = (-1)"; Зп + 2 °л b. b, = 3,…

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Q: 42. Given that la converges, does In (n) 1-1 (n²+5) ² $-10 converge or diverge? A. Dive B. Cannoe…

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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: Set an where r E R is a real number. Prove that ɑn converges to 0 if (-1, 1), cônverges tó 1 if r =…

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

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Q: Consider the sequence S n²r(1 – nx) x € [0, 1/n] - fn(x) := 0. elsewhere . Show it converges…

A: Let gx=n2x1-nx Differentiating wrt x, we get  g'x=n2-2n3x Putting g'x=0, we get  n2-2n3x=0⇒x=12n

Q: If X1, Y1, are two positive unequal numbers and Xn = (xn- 1+ Yn – 1) and y, = Vx, – 1 Ya-1 V n 2 2.…

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Q: State the definition of a sequence xn converting to a point x in a metric space (X, d). prove that…

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Q: ** Consider the sequence (xn) in R, where n (-1)". Show that (rn) doesn't %3D converge to any limit.

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Q: (a) Prove that 4280 k!pk k=0 converges for any r > 0.

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Q: If Ea, converges absolutely, prove that Ea,² converges.

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Q: Given a sequence ( xn ), prove that (xn ) converges to zero if and onlyif the sequence ( |xn| )…

A: From the given information, it is needed to prove that:

Q: the convergence or diver 3k Ek=1 k! 4k Zk=1 k2 - k

A: As per guidelines we have to solve only one

Q: Given f: [2, )IR, prove that fn (x) = けx" It, converger uni formly to f6)-o.

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Q: 2n+1 4. Use the definition to prove that ) converges.

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Q: et {B„} C R be a sequence and x € (0,1). Prove that if |B,+1– B,| 1 hen {B,} is Cauchy and hence…

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Q: 1. If f : [a, b] → R and f(a)f(b) < 0, then f has a root in [a, b]. 2. Every subset of R which is…

A: As per the company rule, we are supposed to solve three sub-parts of a multi-parts problem.…

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

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Q: The sequence a, n – vn+ lvn+3 converges to the limit A) -3 B) -1 C) 1 D) +2 E) -2

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

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Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.