Suppose that a, > 0 and converges. lim na, = 0. Prove that Ea, n 00

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

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Q: D. Use the definition of limit of a sequnce: lim an = L = Ve > 0,3N > 0s t. whenver n> N then Ja, -…

A: We have to prove limit by given limit definition Hence  1. For a sequence 8n2n+3 Consider ε =…

Q: Show that x, converges to x.

A: Given that X be a metric space . Let xn be the sequence in a metric space X. Suppose that xn has a…

Q: Let x1 = 1 and xn+1 1 for n e N. Show that this sequence is convergent and find its In + 2 limit.

A: I have shown the sequence is bounded above and monotone increasing which implies sequence is…

Q: Is the sequen ce in (R.To)? (0,1.0.1.0,1,..) convergent Prove the answer

A: It is given that the topology space ℝ,τ0. Here, τ0=ℝ,∅. That is, the open sets of ℝ  are ℝ and ∅.…

Q: The sequence {( an+2 )"}1 n=1 O diverges converges to Je3 converges to e None converges to Je?

A: the given sequence converges to

Q: Let f and g be as in the figure, we may deduce that if I g(x)dx converges then

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Q: Use the Root Test to determine whether a, converges, where a, is as follows. in n = 1 a in %3D 5"…

A: Here we have, an=n5n By the root test:-

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

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Q: Show that if (Sntn) converges to (s,t) in R² with the do ("max") metric, then the sequence converges…

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Q: Use series to evaluate the limits

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Q: In  the given question as follows , what can you conclude about the convergence or divergence of ∑…

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Q: Suppose that the sequence of functions {fn} and the function f are integrable on [a, b] and that…

A: Given that,  The functions fn and f are integrable on a,b. fn converges to f on uniformly a,b. an…

Q: Let (fn) be a sequence in L that converges uniformly on 2 to f. If µ(2) < ∞ show that…

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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)&gt;14 Taking…

Q: Let X = R' and d be the metric on Rº given by d(x,y) = E., \x - y,| where x = (x, X2, Xa) and y =…

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Q: The sequence ɑn = 1 + converges to the limit A) e B) ve C) Ve D) e? E) F) e? /2

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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: Use the root test to determine whether ye converges, where An is as follows. an =

A: Explanation of the answer is as follows

Q: Let (fn) be a sequence in L that converges uniformly on to f. If µ(2) < ∞ show that Integral (f)…

A: It is given that fn converges uniformly on Ω to f , for ∀ε&gt;0, there is N&gt;0 which is…

Q: Consider the sequence {fn(x)}, where sin(nx) fn(x) = 1+ nx a. Show that fn(x) converges pointwise to…

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Q: Suppose that a, > 0 and lim a, = ∞. Prove that Sa, diverges.

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Q: Use the root test to determine whether . converges, where an is as follows. (1 + Ink)*

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

A: As the mention in above given question when x goes towards infinte than function f(x) gives 0. It…

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

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Q: Prove ca) lim (an t anay t --- + that if Ean Converges, then + antr) = 0, (b) Give o7Convergence…

A: We have to use properties of convergent series first.

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

A: Given a statement

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

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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: State whether the following converge as n → 00; if yes, find the limit. Justify your answer. (а)…

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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: rove that then if Ean converges Lim(ant antit t antr) =0,r20 ハ→ 0 O Does imply that Ean Converges?…

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Q: Let X be a set, and define d(x,y) ={0, if x=y; 1, otherwise Show that, if a sequence (xn) is…

A: Have used the definition of convergence sequence and the discrete metric

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

A: Answer

Q: Show by example that E(a,/b,) may converge to something other than A /B even when A = equals 0. Σαν…

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Q: Suppose an 20 and limnan = L.Then 1. If 0 ≤ L < 1, then Enz1an converges

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Q: State whether the sequence an = Vn +7 converges and, if it does, find the limit. a) O converges to…

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Q: Given that {an}n=1_is convergent such that a1 = V5 and an+1 V5 an- %3D Solve for lim an n 00

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

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Q: If E, 2"an converges and E o(-1)"2"an diverges, then the radius of conver- gence of o anx" is 2.

A: According to the given information, Suppose,

Q: Use series to evaluate the limits

A: Given: To find:  lim t→0  1- cos t - (t22)t4Formula:  cos t…

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: Does the sequence Pn=(120e0.7n)/(4+e0.7n) converge? If so what's the limit?

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Q: O Find the set s on w/c In )= x"(1-x") converges pointwise.

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

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