Let 0 < 0 < 1 be a rational number. Show that the sequence {cos(nT0)}-1 is divergent.

Q: Evaluate the upper sum for f(x) = 1 + cos - ASXST Wwith n = 6.

A: GIVEN: f(x) = 1 + cos (x/2)

Q: Consider the sequence {fn(x)}, where sin(nx) fn(x) = 1+ nx Use the uniform norm to show that the…

A: Given: fn(x)=sinnx1+nx To show: fn(x) is uniformly converge on [1, ∞).

Q: Use series to evaluate the limit. lim 1- cos(5x) x-01+ 5x - e5x

A: Objective is to find the limit

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: Let {xn} be a bounded sequence on R. Show that Xn lim = 0. - n→+o n

A:

Q: Find the limit of the sequence an = In(9n +7) – In n. If the limit does not exist, type "Diverges"…

A: Solution:

Q: Compute the DFT for the sequence [1,1,1,1] x(0)= x(1)= x(2)= x(3)=

A:

Q: Use series to evaluate the limits

A:

Q: Let (fn) be a sequence in L that converges uniformly on 2 to f. If µ(2) < ∞ show that…

A:

Q: State whether the sequence an = sin converges and, if it does, find the limit. a) Oconverges to 1 b)…

A: If f is continuous and {xn} converges to x. Then f{xn} converges to f(x).

Q: Find x so that x, x + 2, and x + 3 are consecutive terms of ageometric sequence.

A: We have to find the value of x such that x, x+2, and x+3 are consecutive terms of a geometric…

Q: 1 n4 + sin“ n=1

A: To find that the given deries diverges or converges

Q: Prove that the sequence xn has a convergent subsequence. 3n4 sin n – 4n cos n2 + 2 a) Xn 5n4 – 8n –…

A: We have to prove that xn has a convergent subsequence in each of the given cases.

Q: Apply the ratio test to the series. n! 2n3 n=1 an+1 Find the limit lim an

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Q: * /In(n), cos?n COS n2 n=| Identify if it converges or diverges

A:

Q: Which of the sequences {an} converge, andwhich diverge? Find the limit of each convergent sequence.…

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Q: 1+ sin(2n) bn V1+n

A:

Q: Evaluate (x2 – 2x³) dx using the limit of Riemann sums.

A: given integration is                                       I=∫01x2-2x3dx we have to find value of…

Q: Suppose that an<0 for all n, and define b,=tan-(a,-n). Then the sequence {b,}... O diverges to - O…

A: Solve the following

Q: Prove that sina de converges but not absolutely for 0 < p < 1. XP

A:

Q: Find x so that x - 1, x, and x + 2 are consecutive terms of ageometric sequence.

A: Given x-1, x, x+2 are the consecutive terms of a geometric sequence. To find x Let t1=x-1t2=xt3=x+2…

Q: an = cos

A:

Q: Determine whether the sequence defined by 2n sin(n) an = 3n2 + 5 is convergent or divergent. If the…

A: Whether the sequence is convergent or divergent by finding it's limit

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

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Q: Use series to evaluate the limit. 4 3 sin(2x) – 2x + lim 4/3

A: Given problem:-

Q: 99. Show that 00 cos x dx is absolutely convergent.

A:

Q: use SERIES to evaluate the limits

A:

Q: find if diverges or converges. if converges, then find its limit cos²(8n) An 8n-1

A: Given - an = cos28n8n - 1 To find - find if diverges or converges. If converges, then find its…

Q: (5) Use sequence theorems to prove that lim0 sin() does not exist and that lim,-0 x sin(-) does…

A:

Q: Prove using the ϵ−n0 definition that the sequence Xn=(9−7n)/(8−13n) converges, and find its limit.

A:

Q: Find the limit of the sequence bn = (cos n)/(n2 + 1).

A: Given sequence is bn=cosnn2+1.

Q: {sin ()} 2

A: Given sinnπ2

Q: Let an be monotone and positive. Assume that an converges. Show that n=1 lim nan 0. n-00

A:

Q: Let {an}=1 be a convergent sequence such that a1 = V5 and an+1 V5 an. Compute lim an.

A:

Q: Let x1 = 1 and define xk+1 = sqrt(2xk) where k is a natural number. Prove that the sequence {xk} for…

A: Given: x1=1, xk+1=2xk So, x2=2x1=2&gt;x1 x3=2x2=22&gt;x2 On generalizing, we get…

Q: Find the limit of the sequence whose terms are given by 4.9 1 Cos In =

A: Let's find.

Q: Find limit of a, of the series (–1)"+1 (, Inn-

A:

Q: Find the limit of the following sequence: 00 cos(2n? + 10) (-1)"-ln3 n=1

A: The objective is to find the limit of the following sequence.

Q: use the appropriate limit laws and theorems to determine the limit of the sequence orshow that it…

A: The given sequence is,

Q: Use series to evaluate the limit. 2 +2 cos t-2 lim t+0

A: Given,           limt→0t2+2cost-2t4

Q: (Xn) be a sequence of real numbers defined recursively by 1 = 1 and 1 7. Let X = In+1 =5(Tn +3) for…

A:

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

A:

Q: The sequence a,=n sin 1 converges Select one: True OFalse

A: We have to check

Q: Let {xn} be a bounded sequence on R. Show that Xn lim = 0. n→+∞ N

A: Given sequence {xn} is bounded we want to show that  limn→+∞xnn=0 we  use the limit limn→∞1n=0

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: If F(x) = 2x2 - 7, 0 ≤x ≤ 3, evaluate the Riemann sum with n = 6

A: We will find Riemann sum for given n and function

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.