EXAMPLE f(z) = Σo(z"/n!) is convergent for all z and, according to Theorem 2.9, n7"-1 8 -n

Q: every convergent cauchy Show that Sequence is but the converse is not true general. in

A: Let, be the convergent sequence which converges to . Since, then, for the given ,

Q: 1 10. Determine by the root test if Ln=1 converges or diverges. (In(n+1))“

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Q: - Find the interval of convergent for : 2n 2 2n! n=0

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Q: 2. Absolute convergent, conditional convergent, or divergent? i) E(-1)k 25k +8 k = 1 ii) E(-1)k 3*+±…

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Q: Suppose that the sequence {n} is such that xn 20 for all n eN and that lim(-1)"xn exists. Show that…

A: We will use Triangle inequality.

Q: (-1)^(n+1)/(sqrt(n)((lnn)^(2)). Prove that those are convergent but not absolutely conve

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Q: 2 ทะ n (8) n! h

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Q: If an with AN>O is Convergent, then is Σ TavaN+1 always convergent?

A: Given,  is convergent, where, So,  is.Now, using A.M≥G.M property,We have,

Q: 3] Let (xn) be bounded (not necessarily convergent) sequence, and assume that lim(yn) = 0. Show that…

A: A sequence is said to be bounded if it is bounded above and below by a number. That is, the sequence…

Q: ㅋ. for neN. Does (xn) have a convergent sub. (gn) have a convergent subsequence? If you are unable…

A: Statement : Every bounded sequence of real numbers has a convergent subsequence.

Q: 2. On X = {u Lip([0, 1]): u(0) = 0, u(1) = 1} and for 20 consider the functional F: X → R Fe(u) =…

A: Given: X=u∈Lip0, 1 : u(0)=0, u(1)=1 ε≥0 Fε : X→ℝ Fε(u)=∫[0,1]1+x+εu'(x)2dx

Q: B){(sin(x))/x

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Q: ii) Prove that if a sequence XnXn converges to a limit l, then any subsequence of XnXnalso converges…

A: Just use definition of convergence.

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: The given sequence is .We need to check that the given sequence is convergent or divergent with the…

Q: Prove that the sequence {f(1/n)} has a limit if |f'(x)| < 1 for all r € (0, 1]. I

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Q: Let Xn=cos пл 2 then limit supremum of the sequence is

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Q: In(1+2e") d) an n

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

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: a) {tan"(n²)} p-1 n=1

A: There are multiple question. As per company policy kindly ask other question separately. a. This…

Q: з Σ (-1)^ hai боu итиеть n ди

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Q: g) an = In(n+ 1) – In(n) (Hint: simplify the logarithms) (-3)" h) an n! i) an = Vn + 2 – V2n

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Q: Exercise 6. Let (an) be the sequence Sao d Ants = 20²0. an +1 Let α = = 42 defined by = and show…

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Q: An = 2n an = 3n – 5 an = (n + 1)/3 an = v2

A: Given:  a) an=2n b) an=3n-5 c) an=n+13 d) an=2

Q: True or false: if (x) converges, then every subsequence (x) is Cauchy. n=] k=1

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Q: bz In Exercises 15-28, use Theorem 1 to determine the limit of the sequence or state that the…

A: The given sequence is

Q: 14. Divergent or convergent? If convergent, show the limit nsin(n) an n2 + 1

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Q: why is no one able to solve this? Can an expert please help

A: Explanation:1. Define zn​ :  We introduce a new sequence  zn​=xn​+yn​ . This sequence represents the…

Q: (-1)"n (xn) = п+2

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Q: e{3k(n +7) sin( 1. -) +7} k(n+7)

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Q: (3) Let {an} be a sequence in R that has a subsequence {an;} satisfying |ani — ani+1| ≥c for some…

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Q: Apply the theorem: /Let (XN) be a sequence of positive real Numbers such that Li 2:= lim (XN+1/XN)…

A: Given sequences: (a): (b): (c): (d): where  and  satisfy We have to find the limit of the given…

Q: Explain why if an converges, then limn→∞ an = limn→∞ an+1. With proof!

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Q: Determine whether it is uniformly convergent £r20 x2+k2

A: Given  ∑K≥0xx2+k2

Q: xplain with the right test if they are convergent or divergent a) n=1 n+ 2n 1+ 3n Σ(n+2)! +2 n=1

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Q: Hi, can you tell me if the following is true or false?

A: First statement is false because