Suppose that the sn satisfies both limn→∞ s2n = 2 and limn→∞ s2n+1 = 2. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 2.) Show that also limn→∞ sn = 2.

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

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Q: Show that the series Ln=l 12+2z10n-5z"ª –3z+2;10n converges absolutely for |리 < 1.

A: The series ∑n=1∞an is absolutely convergent if the series ∑n=1∞an  Ratio test:  The series ∑n=1∞an…

Q: Suppose that the sn satisfies both limn→∞s2n = -1 and limn→∞s2n+1 = -1. (That is, the sequence given…

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Q: Suppose that a power series f(x) = L a„x" n=0 has the radius of convergence p > 0 and that L.- anp"…

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Q: Suppose that we have two series with positive terms, =1 n and En bn- Then we may apply the limit…

A: Suppose that we have two series with positive terms Σan and Σbn .

Q: 2. Show that if a > 0, then the sequence (²x²e¬n) converges uniformly on the interval [a, ∞), but…

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Q: In(n + 9) State whether the sequence ɑn = converges and, if it does, find the limit. n а) converges…

A: Given data is : an=lnn+9n

Q: * Show that the sequence (xn)n>1 in R, where In = 1- Vn' converges to 1.

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Q: 20. Show that if E-1a, converges, then 1 + sin(a,) Σ n=1 converges.

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Q: 00 Fn+1 { is convergent, and Show that the sequence Fn n=1 1 Fn+1 lim n→∞ Fn Fn+1 > 1 and Fn-1…

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Q: Suppose {fn}=1 and {g„}=1 converges uniformly on E where {fn}=1 and {gn}n=1 are sequences of bounded…

A: Let fnn=1∞ is converges uniformly to f and gnn=1∞ is converges uniformly to g on E and fnn=1∞ and…

Q: The sequence (n+(-1)" n Diverges Converges to 0 Converges to 1 Converges to e Converges to 2e

A: Option 3 is correct.

Q: +o0 (-1)*n" sin" () Σ 2n n=1

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Q: Define the sequence {Sn} by Sn= [ 1 / (2)(1) ] + [ 1 / (3)(2) ] +   .............  + [ 1 / (n+1)(n)…

A: The nth term of the partial sum is given by

Q: Determine if the sequence converges or diverges. If it converges, state the limit. cos²n an = -,n≥1…

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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: Suppose that a, 2 0 and b, 2 0 for all n> N where N is an integer. If an lim = 00, what can you…

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

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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: Compute the limit of the nth term and use it to explain why the series is divergent. +00 2" – 1 2 2n…

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Q: 5. Use Theorem 5.10 (Section 5.3 in Vol. 2 of OpenStax Calculus) for this problem. How many terms of…

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Q: If the lim, (1+)" =e, then the series a, is (i) Converge (i) Diverge (ii) Inconclusive (1)

A: we have given that limn→∞1+1nn=e

Q: Show that the series n" n' is convergent. Deduce that lim (2n)! (2n)! n=1

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Q: 4. If (x) is a sequence in an inner product space X such that the series converges, show that (sn)…

A: It is given that xj is a sequence in an inner product space X such that the series x1+x2+⋯…

Q: Show that the series n' (2n)! is convergent. Deduce that lim (2n)! n=1

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Q: oll Show that sequence {n} n = 1, where .... 1/1₂ 0₂0₂...) converges 1/1,0,0 :) 9 9... 1/ng...) in…

A: We have to show that the sequence xnn≥1 defined by ; xn = 1, 12, ..., 1n, 0, 0,...  converges to  x…

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

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

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Q: converges for all * > 0 because lim a - In n 0, for all * > 0- n=2 True False

A: Need to check given statement is true or false.

Q: Show that the series 71 n' (2n)! is convergent. Deduce that lim 0% (2n)! n=1

A: We have to show that the series ∑n=1∞nn2n! is convergent. Let an=nn2n! Now,…

Q: 8. Use the fact that ()1 converges to 0 along with the theory derived in this section and worksheet…

A: We can solve this by the help of squeeze theorem and given details

Q: Let 2 ak converge with the positive Ak Converge with the positive term- In this case examine. the…

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Q: { { In(n) 5n² Sn=1 Determine whether the sequence converges or diverges. If it converges, compute…

A: We need to check whether the sequence lnn5n2 converges or diverges.

Q: 2. Use the definition to prove that if (xn) is a sequence of real numbers which converges to -3,…

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

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

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Q: The sequence (Xn) = (1, 3, 1, 3, 1, . . . , 1, 3, ...) is: Converge to 3. O Divergent. Converge to 0…

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Q: Use Taylor series to evaluate the limit. lim x -4 -1 O 4

A: Given: limx→∞x2e4x2-1 ....(1) From Taylor's series we have e4x2=1+4x2+124x22+164x23+...…

Q: Use the Root Test to determine whether the series convergent or divergent. 2n -4n n + 1 n = 2…

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Q: 4. Show that the sequence { fn}, where nx fa(x) = xER 1+n°x2 is point-wise convergent but is not…

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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: 8. Suppose ar > 0 and Does > ak converge or k diverge? Why? %3D

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Q: Let fn(x) be the following sequence (not series) of functions n2 fn(x) = (x2 + 7x + 6)n² +n

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

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Q: Suppose that {xn} is a convergent sequence and {yn} is a sequence such that for any e > 0 there…

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Q: 2" Зп+2 n=0

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Q: 5. Show that the sequence Pn = 1 converges linearly to p 0, and the sequence Pn = 10-2" converges…

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

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

Q: Show that the series n" (2n)! is convergent. Deduce that lim 0. (2n)! n=1

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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.