6. Suppose a, converges. Assume the terms a, are non-zero. Prove ÷ diverges if it is true. n=1 n=1 If it is not true, provide an example to illustrate thìs.

Q: Ex. Show the seguence whose nth term is In (n*) an %3D converges. 5n

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Q: If 2a, converges, then 1. diverges Select one: True False

A: Use the defination of convergence of a series.

Q: What is true for the sequence of general terma, =1+(-1)" ? Select one: O converges to 2 O None of…

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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: 1. Use an appropriate way to determine whether the following is convergent or divergent. k 'Ink i. Σ…

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Q: Determine whether the given converges or diverges. n Σ 2n + 1 n=1

A: Given is a series ∑n=1∞n2n+1n To check if the given series is convergent or divergent Ratio test:…

Q: Determine whether the following converges or diverges {(5) n 2n +1 n=1

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Q: 8. Determine by the ratio test if 2n=1 (2n)! converges or diverges.

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Q: converges, because 0 < 1 and an %3D debook 2n + 1 2n +1 2n +1 Room

A: We have to choose the true or false

Q: 52n 5n +1 n=1

A: Given that:- ∑n=1∞52n5n+1To find:- chek whather the series is convergent or divergentSolution:-limit…

Q: N! 9) Show that converges/diverges

A: Use D Alembert ratio test

Q: Show that > iP converges to a finite value iff p < –1. i=1

A:

Q: Determine whether En=1n converges or m%3D1 2n

A: Given: ∑n=1∞n2n

Q: What is the largest positive constant K such that if 0 <r < K, then (nl)2 must (2n)! n=1 converge?

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Q: Show that if an < o0, then E In(1+an) converges if and only if an converges. n=1 n=1 n=1 Hint: show…

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Q: The series 7 2n E,(-1)" 4 n= is absolutely convergent. Select one: True O False

A: Root Test : Suppose we have series ∑ an .Define,                          L = limn→∞ an1n .Then ,(…

Q: If a, converges and a, > 0 for every n, show that 4converges. What can you say about a for any…

A: Given: ∑n=1∞an converges and an&gt;0 To prove: ∑n=1∞an2 converges. Proof: ∑n=1∞an2=∑n=1∞an2 Since…

Q: (3n)" (2n + 50)" n=1

A:

Q: Consider the sequence n²r(1 – nx) x€ (0, 1/n] =: (x)"f elsewhere . Show it converges uniformly on…

A: Consider the sequence  fnx=n2x1-nx      ,x∈0,1n0                       ,elsewhere The limit function…

Q: O Determine whether En=1 st. converges or diverges. Justify your answer by citing a 3n+2

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Q: If 2 a, converges, then la,| diverges. n=1 n=1 True False

A: Given statement is FALSE. I HAVE given a counter -example.

Q: If 2, =1ª, converges conditionally then E=|92| diverges 'n=D: Select one: True False

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Q: Determine whether the given converges or diverges. n 2n +1 n=1

A: We can solve this as follows:

Q: H.w. which of the fellewing serier absolutely converge and which eonditionally convere.

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Q: If an < bn and if En=1 an converges, then 1 bn also converges. True False

A: we can solve it by using comparison test.

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

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Q: Determine whether E=1(-1)"+1. 2n converges or diverges. If it converges, find the sum. 3n

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Q: If 0< by <1 for all k and , br converges, then , converges. Lk-1 %3D1 Select one: True O b. None of…

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Q: (2) If £ an and D bn are both divergent then is E an · bn necessarily divergent? If yes, provide a…

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Q: 4. Let r be a positive number. Determine the value(s) of r, if any, such that converges. As n" n=1…

A: Explanation of the answer is as follows

Q: d) If Can's Converger and {b,'s diverges, Hhen fanton} Converges.

A: we argue by contradiction: suppose that an+bn converges. Since an converges by assumption, the…

Q: The sequence {(1 + 1)^² +1} 2n O O converges to e² is divergent ez converges to e 2 O converges to e

A: Sequence

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: Vn +1- n –1 n=1

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Q: (-1)" In(n) Vn n=2

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Q: 1 7. Determine by the ratio test if n- converges or diverges. п! -

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Q: ,(-1)" converges conditionally, but not absolutely. Show all work. 3n Prove E

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Q: The sequence Converges to e Diverges Converges to 2e Converges to 1 Converges to 0

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Q: 9. Use the definition of convergence to prove that the sequence converges. b) {2n/(3n + 2)} c)…

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Q: By examining the limit as n→0, the sequence {2"}D=1 O Diverges O Converges to 0 1 Converges to

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Q: Given that " In (1) converges, does [.. converge or diverge? In(n) 1 (²+5)² A. Diverges B. Cannot be…

A: Let ∑pn and ∑qn are two series such that pn≤qn. If ∑qn is convergent, then ∑pn is convergent.

Q: Suppose {fn}n=1 converges uniformly on E. Show {fn}n=1 converges pointwise on E.

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Q: 1dx x²+2x+5 -00 O diverges O None 37 converges to 4 converges to converges to converges to 3r

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Q: Write the first five terms of p_j= 1/j and then prove p_j converges to 0. explicitly write the…

A: Concept:

Q: If E=19, converges conditionally then E,-|a,| diverges Select one: True False

A: If absolut series is divergent than alternate series is convergent.

Q: O Does Eo 2k+² e-k converge or diverge? If it converges find the sum. "k=1'

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Q: Does E 8. k=1 In(ek41) converge or diverge?

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Q: 3k 5- 2k+1 k=1 2 1. Does converge? If so, what does it converge to? If not, justify.

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Q: 6n2 n

A: The given sequence be 6n2en To find it is convergent or divergent