Σ(1 is n n=1 convergent divergent Check one n Reason: lim In 1 lim n In 1 n 1 1 1. lim 1 2 n 3 n2 4 n3 1 Hence lim 1 + 0.

Q: 2n +1 a) Using the definition, show that lim %3D n-00 3n +4 3 b) Show that the sequence {a,} defined…

A: As per our guidelines we are supposed to do only first question among multiple questions. To get the…

Q: n sin n Determine whether En-1(-1)". n3 is absolute convergent, conditionally convergent, or…

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Q: Using the nth term test for divergence, the series 00 00 3n an n=1 n=1 is divergent since lim an= 10…

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Q: (6) Let {an} be a sequence of positive numbers. Define sequence {xn} in R² by (sin am Va +1 an An +1…

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Q: 1 1 Зп — 1 3n + 2 n=1

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Q: Let a, = 2n+1– 2". Then, lim an %3D n-00 None of the other options the above co and the sequence…

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Q: n k In- 1 Show that In bn k=1 Show that Inb, converges to In x dx. (Hint: Consider a Riemann sum.)…

A: Given, bn=(n!)1/nn a)  RHS = 1n∑k=1nlnkn=1n[ln…

Q: nl n-1 lim O converges O diverges

A: Given query is to find series converges or diverges.

Q: (a) Show that $\Sigma_{n=0}^{\infty} x^{n} / n !$ converges for all $x$(b) Deduce that $\lim _{n…

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Q: +o0 (-1)" n (a) ► n! n=1 1 (b) E -3 n - -n n=1

A: Find the limit of the given functions at infinity

Q: e. True or False The Limit Comparison test is conclusive if L= lim 2= 1. %3D f. True or False <1 for…

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Q: السؤال 6 If lim=c>0 , then b, . a, both convergent and both divergent. ylgn O ihi O

A: True

Q: 2n2 + 3 Let Sn ak. If Sn for each n > 1, which of the following statements is n2 +1 k=1 correct? (a)…

A: Given       Sn=∑k=1nakan=2n2+3n2+1  for each n≥1

Q: QUTest for convergent and divergent : (show your work) e-2 1. Σ1-1 n+cos²n 9n 2. En=1 (-2)n+1.n

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Q: an+1 (a) lim = 9 n- 00 an O absolutely convergent O conditionally convergent divergent cannot be…

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Q: 6n + 2 6n+2 Let > and an 5" 5" n=1 an+1 a) Find lim n-0 an 6n+2 b) Determine whether 2 is convergent…

A: Convergence of series.

Q: Q(A). Let {fn(x)} defined over [2,3] Show that (a) fn(x) is meaurable and monotonic increasing for…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: The n-th term in a sequence an is given by an = 3-eol (a) Is this sequence increasing, decreasing or…

A: Given:           an = 3-e-3n (a) Here:      a1 = 3-e-3(1) =3-e-3        a2 = 3-e-3(2) =3-e-6…

Q: The n-th term in a sequence an is given by an = 4 - ea. (a) Is this sequence increasing, decreasing…

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Q: Let a, = vn +1– yn. Then, lim a, = None of the other options -00 and the sequence (a,,} is divergent…

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Q: fa (z) = Question 8 For n 21 and r > 0, let 1 fn (x) = - (a) Show that lim, fn (x) = 0 for r> 0. (b)…

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Q: Theorem? The bisection approach is always Convergent proof we have /*<{| |- 의 also osk-ek 16-9 K |…

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Q: QUICK CHECK 5 Explain why E 0.24 k=0 converges and why E 2* diverges. k=0

A: Given- ∑k=0∞0.2k and ∑k=0∞2k To explain- why ∑k=0∞0.2k  converges and why ∑k=0∞2k diverges. Concept…

Q: the series %3D an V52n+1 n=1 n=1 is divergent since lim an= %3D n- +o 16 This option O This option…

A: 5^lim(2n+1)/(n)=5^2=25.

Q: cos(nt) Σ n = 1 1 Select one: a. divergent by comparison test b. convergent absolutely c. convergent…

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Q: Q.22) The integral x-3 dx is convergent if the values of p are: a) (-0,3) b) (-∞, -2) c) (3,00) d)…

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Q: n!n! is absolutely convergent using the Ratio Test. (2n)! 2. Prove n=1

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Q: 00 n+4 Consider the series a, where a, = + n=1 Compute the limit: lim yan = (Enter "infinity" or…

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Q: f(n) f(n) +1 2. Let f : [0, +0) → R be such that f(x) > 0 for all . Define an Show that the sequence…

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Q: 2 (n2) 00 Show that > diverges. Recall from the Laws of Exponents that 2 n! (n²) = (2")". n= 1 .....…

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Q: 1 2аn + 3 1 for all n > 1. Show that the sequence {an} is 8. Let a1 > 0 and an+1 = convergent and…

A: We will find out the required solution.

Q: k ) , which statement is true for series =1 Uk = Ek=1 Uk uk+1 2k If ρ -lim

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Q: 6n Let an 4n + 1 (a) Determine whether {a,} is convergent. O convergent O divergent 00 (b) Determine…

A: use the basic property of sequence and series

Q: What can you conclude about the convergence or divergence of an? (converges, diverges, or…

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Q: (1) E (-1)*cn must converge if c, > 0 and lim Cn = 0. (ii) E ön must converge if 0 0 and lim 5. 0…

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Q: 8. True or False: If lim an = 0, then an converges. %3D n=0 (a) True (b) False

A: Solved below

Q: 00 Consider the series (-2)" Find the limit an+1 lim (2n-1)! an n=1 n- 00

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Q: 1 diverges if it is true. An 6. Suppose An converges. Assume the terms an are non-zero. Prove n=1…

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Q: 00 Σ 6. a, where a, = sin" Consider the series n=1 Compute the limit: lim yan = (Enter "infinity" or…

A: ∑n=1∞ an here we have an=sinn6n3 use the square root test L=limn→∞an1n=limn→∞ann if L&lt;1 then the…

Q: n! converges. Hint: Use lim Show that = e. пh п-1

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Q: n3 + 5n 8n2 + 5n + 1 4- n=1 (A) diverge (B) converge 1 5- (A) diverge (B) converge n( Inn)² n=2

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Q: If 0 1, which of the following must be true? A) If lim an = 0, then > bn converges. в If an…

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Q: . The sequence converges to zero; The sequence diverges to infinity; The sequence has a finite…

A: here (F) converges to zero are 3,5 (I) diverges to infinite are 2 ,4,6,7 (F) the sequence have…

Q: etermine if the following infinite product converges or II (1 e. Ïi (1+n 12) (-1)" n=1 00 II…

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Q: 5n Let an %3D 4n + 1 (a) Determine whether {a,} is convergent. convergent O divergent 00 (b)…

A: Given -  an = 5n4n + 1We have to determine-(1) Whether an is convergent or divergent.(2) Whether ∑n…

Q: 2. Suppose a1 > 0 and an+1 = an + 1. Prove that {an} diverges to o. Hint: Assume an An converges to…

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Q: n" + sin(18n + 14) n18 + 14 Determine whether the sequence an = converges or diverges. If it…

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Q: 1 Σ 2n + 1) ! Find the radius of convergence for n=0 1 Solution: Set Then %3D 1)! 2n + 1 |a,| 1…

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Q: f(2n + 1). im+o f (x) exists, if {an} = f(2n) and {b,} = f(2n + 1) ? Explain. an} converge? If so,…

A: Given that, f(x) = cos (x pi/2) It is also given that, a_n = f(2n) a_n = cos (n pi)  a_n = (-1)^n…

Q: 65. Let S = E cn! where c is a constant. Σ n" n=1 (a) Prove that S converges absolutely if |e| e.…

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