Let 0 for all n EN. If (x,) is convergent, then lim (xn)=5 4 Select one: O True O False

Q: (b) Let a1 = 1 and an+1 = lim, an- /am+2. Prove that a, is convergent and find %3D

A: The recursive relation represent a bounded above increasing sequence

Q: If lim an 1 then n00 Σ an n=1 converges O diverges O may or may not converge

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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: Determine whether the statement is true or false. If lim a,= 0, then a, is convergent. n- 00 O True…

A: Take the series as an example. ∑n=1∞1n For which an=1n and limn→∞an=limn→∞1n=0

Q: Show that if Σ i=1 converges towards a € R , then lim ) a; = 0 i=N

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Q: (d) Let x* = 1/2(2*), and what is the limit of x* as k 0? What is the order of converge for x?

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

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

A: True

Q: B. Prove that limx → c f(x) = o if and only if for every sequence aŋ that is in the domain of f and…

A: Consider a function fx. Suppose that limx→cf(x)=∞. That is for all N>0, there exists a δ>0,…

Q: Use series to evaluate the limits

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Q: Let 0 for all n EN. If (x,) is convergent, then lim (x,) = 2 4 Select one: O True O False

A: If xn is convergent, let us assume limn→∞xn=x Also, limn→∞xn+1=x Given, xn(1-xn+1)>14 Taking…

Q: Sequences {xn} and {yn} such that they are both convergent and {xn+yn} maps to 5

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Q: In the given question as follows, determine whether the statement is true or false. If it is false,…

A: Given statement, If limn→∞ an=0  then ∑n=1∞an converges

Q: For each nN, consider x,, H (a) (b) 71 X with x E R. Define the sequence {gn} by 8n(x) = f(x) = 1…

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Q: an Show that if lim =0 and E-i b, converges then En=1 converges too. an =1 n+ 00

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

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: 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: If lim a, is not 0 then 2 a, is convergent n=1

A: By using theorem we can say that

Q: an If lim Σ =0 and En=1"n converges, then bn 'n=1 b, diverges.? n+ o0 True O False

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Q: (a) Prove that (2x + 2) > 1 for all x > 0. (b) Let so > 0, and sn+1= } (2s, + ) for all n > 0. Prove…

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Q: V. Let f(x) be a differentiable function such that for all æ > 1, • f(x) > 1, • f'(x) < f(x) and lim…

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Q: Show that if $a_{n}&gt;0$ and $\Sigma a_{n}$ is convergent, then $\Sigma \ln \left(1+a_{n}\right)$…

A: Here, we are given that an &gt; 0 and series is convergent. We need to prove that: is convergent.…

Q: Use the Ratio Test to determine whether an 15" n! Evaluate the following limit. an+1 lim n→∞ an an…

A: This question can be solved using the concept of test for convergence.

Q: If lim,f(x) = 0, then f(x)dx converges Select one: O True False

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

A: Given a statement

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: +9 for all n2 1. Prove that {r,} converges and find 2.1 4. (a) Let r = 4 and am+1 its limit.

A: Sometimes, a sequence is defined recursively. In that case, to test the convergence of the sequence,…

Q: The sequence {an} is defined by a1 = : 2, and An + An An+1 = for n > 1. Assuming that {an}…

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Q: Let sn = a1 +.. + an. If lim s, = 0, then the series an is convergent. %3D ...

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Q: Let a 1 and define an+1 = V4an -I for n2 1. Show that the sequence {an} converges and find the…

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Q: V. Let f(x) be a differentiable function such that for all æ > 1, • f(x) > 1, • f'(x) < f(x) and lim…

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

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Q: Evaluate An+1 lim for n! An nn and determine the serie Σ n=1 converge or diverge? 1/e and diverge e…

A: We have to check

Q: If lim=c>0, then both convergent and both divergent. ylgn O Ihi O

A: If limn→∞anbn=c&gt;0, then ∑bn,  ∑an both convergent and both divergent. (a) True (b) false

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: If fn If 0 ≤ an ≤ bn for all n € N, and f on [a, b], then lim ſå fn ‡ få f. an converges, then bn…

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Q: k en(k) 大=Z

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

A: Answer

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: ca, is convergent. is an convergent and c is a nonzero constant, then If n=1 n=1

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Q: If lim,→00 an 0 then an converges. Select one: O True False

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Q: a) If lim an = 0, then Σ an converges. n→1

A: The divergence test states that for a series ∑n=1∞an, if limn→∞an ≠ 0 , then ∑n=1∞an diverges . That…

Q: Use series to evaluate the limit. V1+x-1 1 ーX lim

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Q: dx heorem 1.7.1 is convergent if and only if µ < 1. (x – a)H a

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Q: Σ 1. 4n n=1

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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: Show that: For any f E C(R/Z; C), the If (n)|² is absolutely convergent, and n=-00

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Q: 7. Suppose (fn) is a sequence of continuous functions on an interval I that converges uniformly on I…

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Q: converges, what is its limit? 2) ) (In(1+ 4n)) In(n + n=1

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Q: ar converges if and only if lim a, = 0. r->0 ア=1 True False

A:

Q: find if diverges or converges. if converges, then find its limit cos² (8n) an where n 2 0 8n-1

A:

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.