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

Q: Show that the sequence {f}, where f₁²x) = tan `nx, x70 is uniformly ノ on any interval [a,b], aso but…

A: Given function as fnx=tan-1nx, x≥0

Q: Determine the given below serie is converge conditionally, absolutely converge or diverge 4n° + 4 8n…

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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: Prove or provide a counterexample for the following: a] If (s,) converges then (s,) converges.

A: We will prove that if { sn } converges then {|sn|} also converges.

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: 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: 00 Show that converges if [r| < 1. п: п

A: The given series is

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

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Q: Prove that if a sequence  {an} converges, it must be bounded. Is the converse true as well

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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: Prove that {x_n} converges. Need to use info in a) and b)

A: We have the following information from part(a) and part(b) for any a,b∈ℝ…

Q: is it seies converge ost not. 1-2)^

A: To find:

Q: If C.)" converges, then C,)" converges. n=0 n=0 Select one: True False

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Q: Prove: if 0 is less than or equal to r which is less than or equal to 1, then (r^n) converges.…

A: The objective is to show that  if 0≤r≤1 then rn converges. For n=1,  0≤r≤1 So, r bounded. For n=1 it…

Q: Suppose a,>0 for all n and 2 a, converges , which of the following must converge n = 1

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Q: What are all of the values of x ,such that - (-1) n+A converges? n=1 .1 X26 .2 06

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Q: (b) Give an example to show that the product (fngn) may not converge uniformly.

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Q: Prove that if Ean converges , then (antanti+ --- anor)=0, Does it imply that Ean converges? Give…

A: Given: to prove, limn→∞an+an+1+.....an+r=0,r≥0

Q: If X1, Y1, are two positive unequal numbers and Xn = (xn– 1+ Yn – 1) and y,= Vx – 1 Yn –1 V n 2 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: Suppose Σan and Σ bn converge where an and bn are nonnegative for all n. Prove that anbn converges.

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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: If an > 0 and an converges, then (-1)"an converges. Select one: O True O False

A: Absolute Convergence test :The absolute convergence test states that, If ∑n=0∞anconverges then,…

Q: H) IM:

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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: If Ea, converges , then n=1a, diverges n=1 an n=1 Select one: O True O False

A: given sequence are, given that 1st sequence is converges and we have to find 2nd sequence is…

Q: @ suppose {an} and { bn} are sequences such that {an} and { ant bn'} converge.prove that {bn}…

A: To prove (a)  Given that an converge and an+bn converge. The objective is to show that bn converge.…

Q: Use only Definition 3.1.1 (or its negation) to prove the following: 2. Prove that the sequence…

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Q: Prove that 「+ X,- ズー 2. 71 ,nyo converged number J2 ヒo

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Q: Using Sandwich Theorem, the sequence 2 1. an = 4 sin? 3 %3D n + 1 converges to 0 converges to 2…

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Q: Prove that if 0 ≤ r ≤ 1 then (r^n) converges. Then prove that it converges to ???

A: We need to prove that |rn| converges for 0≤r&lt;1.

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

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Q: 1/n 72,

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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: I En=1 an converges then 2a, converges.

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Q: If > an converges, then a converges. n=1 n=1

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Q: Set an where r E R is a real number. Prove that ɑn converges to 0 if (-1, 1), cônverges tó 1 if r =…

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

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Q: Does a(n)= n^2sin(3/n^2) converge?  If so, to where

A: The given question can be solved as shown in step2.

Q: Suppose {a,} is monotonic. Prove that {an} converges if and only if {a} converges

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Q: Given f: [2, )IR, prove that fn (x) = けx" It, converger uni formly to f6)-o.

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Q: . Prove that (-Sn+7ys ,5n²-5n+7, 3n²+4n-8' 00 n=1 converges.

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

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Q: Prove if ( a n ) converges, then ( | a n | ) converges.

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Q: Using the same proof that we saw in class that shows that n-1 7 converges, prove that n=1 na…

A: Given the series ∑n=1∞1n2 converges. To show that the series ∑n=1∞1nα converges when α&gt;1.