In each case Stabe whether the Statement is true or false, Explain your answerseo with Show'ing examples. a) If the tems of Convergent Se quence are all Posittive, then Lim, nう an is Positive. b) If lim an=1, then a, + a, +92 +-- is unbounded. cl If [an} diverges and {bn} duerges, Hhen {antb,7 diverges.

Q: |an also converges because 0 If an converges then < lan|, for n = 1, 2, ... n=1 n=1

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Q: Prove the limit below exists and find its value to 6 digits but replacing n for a large value and…

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Q: Using the limit definition, prove that if {an} converges and {bn} diverges, then {an + bn} diverges.

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Q: Prove that if the sequence {an} is divergent and the sequence {bn} is convergent, then the sequence…

A: Given: The objective is to prove that if the sequence {an} is divergent and the seqence {bn} is…

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

A: Use the defination of convergence of a series.

Q: Discuss the Conditions foo which Foudies Segies is convergent?

A: Given: Conditions of Fourier series is convergent.

Q: A sequence (xn)∞ {n=1} is said to oscillate if lim inf xn < lim sup xn.  Prove or disprove the…

A: We can solve this question using definition of converges diverges and oscillating of sequences

Q: Prove that the sequence given by 2.4. 6.(2n) Un = 3.5.7.…(2n + 1) converges and find its limit.

A: Given query is to prove that sequence converges.

Q: Question A.) Does this CONVERGE or DIVERGE? er 2 (lan) ² W nal

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Q: If 2 a, is convergent with positive terms n=1 is it true that sin (a,) is also n=1 convergent?…

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Q: . If [an is a senies of pobitive terms md [an is convergent then cheek the convergenie of the…

A: Comparison test : If 0≤yn≤xn for n≥k for some k .Then  If ∑n≥1xn is convergent then ∑n≥1yn is also…

Q: If lim bn pe 0= =1 || En=1 a, converges, then En=1 b, diverges.? True False

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Q: 7N 8N 3) Show that converges/diverges.

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Q: 27 81 2) 16 – 12 + 9 – + 4 16 243 + 64 - - -

A: Given the series: 16-12+9-274+8116-24364+............ To check if the given series is convergent or…

Q: Suppose you run a limit comparison test and determine lanl diverges. In your written work: Nothing,…

A: If series of absolute term diverge than series doesnot converge absolutely.

Q: Show by example that Σ(an/bn) may diverge even though Σan and Σbn converge and no bn equals 0.

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Q: If lim =c>0 , then b, , a, both convergent and both divergent.

A: By limit ratio test we know that if above condition is true then either both series diverges and…

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

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Q: We can show that the series 1 2" – n n=1 Σ is convergent by applying the limit comparison test with…

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Q: Suppose that (sn) and (tn) are sequences so that sn = tn except for finitely many values of n. Using…

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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: Suppose that the sn satisfies both limn→∞ s2n = 2 and limn→∞ s2n+1 = 2. (That is, the sequence given…

A: Step:-1 Limit of a sequence:- Let f(n) be a sequence of real numbers and let l be a real number then…

Q: Show that every convergent seapuence is Cauchy but the convesse is not true in general.

A: A sequence is said to be Cauchy if there exist a natural number N and a small positive number such…

Q: Show that if lim 0 and E- b, converges then Ena, a, converges too.

A: We have to Prove that series an is convergent.

Q: show work to prove if the serie is convergent or divergent?   as n=&gt; infinity   an= (ln(3n)) /…

A: The given series is,

Q: a, If lim=c>0 , then b, 2 a, both convergent and both divergent. ylgn O İhi

A: Given a statement

Q: of Eca 4 is cenvengend can we conclude that cach of these are convergent: (b) E cn (-4)^ ca)

A: Given that ∑n=0∞Cn4n is convergent.

Q: If limkoo ak 0 then Eak is convergent.

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Q: Suppose that a, 20 and b, 2 0 for alln> N where N is an integer. If an lim = 0, what can you…

A: Consider the given: Limit Comparison test:an,bn&gt;0  ∀ n≥Nlimn→∞anbn&gt;0,∑n=1∞an converges then…

Q: show work to prove if the serie is convergent or divergent? an as n=&gt; infinity, ((-1)n (3n)) /…

A: To determine that whether the given series is convergent or divergent.

Q: 5 If 2 a, is convergent, then lim a, = 0. n=1 Select one: True False

A: We have to show that given statement is true or false.

Q: 28. yn = ne/n

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Q: Example: Show by contradiction that the sequence (un)n = (cos n)n is divergent. Assume that (un)n is…

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Q: 2) In a, = (1+-)" {a,}is convergent. show that the %3D n

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Q: Let an and bn be infinite serics with positive terms. Which is a conclusion of tl n=1 n=1 Limit…

A: option (a) is correctsee below the clarification

Q: If Ea, converges absolutely, prove that Ea,² converges.

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Q: chooes led R- to ) and P= then OR convergentan P divergent O both Rand Par convergent O R…

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Q: converges, because lim n! 2(n+1) n! = 0. (n + 1)! 2* n+00 True False Reset Selection

A: The given statement is true. Use the ratio test to find the evaluate the nth term.…

Q: If 21ak converges then E,ak, converges for any ak subsequence of ak.

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Q: If 2(-1)" a, is absolutely convergente the n=0 na, is divergente. Select one: O True O False

A: Topic:- sequence and series

Q: 4. If (1,) is Cauchy and has a convergent subsequnece (r), then prove that (r,) is convergent and…

A: Given that the sequence xn is a Cauchy sequence and it has a convergent subsequence xnk

Q: Show that Seauence Converse is not tade in gehQlali every Convergent is čauchy But

A: We need to prove that every Convergent sequence is Cauchy but the converse is not true

Q: an = 0 and bn Σ-1 bp Show that if lim n converges then En-1an converges too.

A: Given that:            limn→∞anbn=0 and ∑n=1∞bn converges. Prove ∑n=1∞an converges.

Q: Determine whether the sequence an = arctan(12n") converges or diverges. If it conver Converges…

A: Introduction: We know that the series an is converging if the limit of the sequence an as n tends to…

Q: (4) Suppose {a„}1 converges to L and that a, 2 0 for all n. Prove that {Van}1 converges to L. Tip:…

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Q: 2" Зп+2 n=0

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Q: 17n + 3 Determine whether the sequence an converges or diverges. If it converges, find the limit.…

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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.