(b) Define B C e? by B = {x € l² : E, nlæn[² < 1}. Show that B is compact.

Q: 3 Evaluate :/ (x +3) dx using the limit of the Riemann sum with n partitions using right endpoint…

A: The integral is given as, ∫13x2+3dx Here, a=1b=3fx=x2+3 According to the Reimann Sum Method,…

Q: Let (h,), (tn) be sequences of bounded functions on A that converges uniformly on A to h, t…

A: Given that there are two sequence (hn) and (tn) be sequence of bounded function on A that converges…

Q: Calculate the midpoint Riemann sum for f(x) = Væ on [3, 5]; n = 4

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Q: Write Va dx as the limit of Riemann sums with equal length subintervals using RIGHT ENDPOINTS as…

A:

Q: Let f be a uniformly continuous function on (a, b) and suppose that {xn} is a Cauchy sequence in…

A:

Q: Use the Integral Test to determine the convergence or divergence 1 n5 n = 1 xp converges diverges

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Q: Calculate the midpoint Riemann sum. f(x) = , +1 on [2,7]; n = 5

A: In the question we have to calculate mid point Riemann sum. As asked by student we have to find part…

Q: Let f(x) = (ncosx)/(n+ex) x in [0,1], n in N Prove that the sequence converges uniformly on [0,1

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Q: Let f(x) = (ncosx)/(n+ex) x in [0,1], n in N Prove that the sequence converges uniformly on [0,1].

A:

Q: Use the Bolzano-Weierstrass Theorem to show that (Xn) = (1, 3, 5, 7, 1, 3, 5, 7, 1, 3,5, 7, . ....)…

A:

Q: B) If the partial sums s, of Ea, are bounded. Show that converges to n(n+1)

A: Let the partial sums Sn of ∑1∞an be bounded —iConsider,      ∑n=1N Snnn+1 = ∑n=1N∑k=1Naknn+1…

Q: 4. Let f: DCR→ R be uniformly continuous and {xn} be a Cauchy sequence in D. Then show that {f(Xn)}…

A:

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

A:

Q: Let fn(x) = n²x²(1 – x) on [0, 1]. On what closed subset, F C [0, 1], does the sequence {fn}…

A:

Q: where Oso, using the factori Zation theorem, Tn = #x;, is a sufficient stati Show That for 0.

A:

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: to Find Fxample Sum t W o subrings?

A:

Q: Estimate the area under the curve y=(x^2)+1 between x=0 and x=1 using Reimann Sums with four…

A: Given, y=x2+1 a,b=0,1 n=4 ∆x=b-an=1-04=14 The four intervals are, 0,14,14,24,2434,34,1 The right…

Q: Show that an abbolately converget sover ss mvergentserer es Condegent but Converse es not true în…

A: To show that if a series is absolutely convergent than it is convergent but converse need not be…

Q: Find the Riemann sum for f(x) = x + x², –2 < x< 0, if the partition points are -2, – 1.5, –1, -0.7,…

A:

Q: Prove that the sum T in the Trapezoidal Rule for ∫ba ƒ(x) dx is aRiemann sum for ƒ continuous on 3a,…

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Let (hn),(tn) be sequences of bounded functions on A that converges uniformly on A to h,t…

A:

Q: the sequence {fn} converges uniformly to f on R. Prove that the limit function - R be bounded.…

A:

Q: 3. Show that the ratio test is inconclusive, but use either the integral or a comparison test to…

A: Given the series:

Q: 4. Let f : DCR → R be uniformly continuous and {In} be a Cauchy sequence in D. Then show that {f(1)}…

A: Image of a Cauchy sequence

Q: It can be shown that lim beb = 0. Use this fact and the integral test to show that k is convergent.

A: Given infinite series is: ∑k=1∞kek Integral test states that the given infinite series will be…

Q: 1 Use the Cauchy's integral test to show that Σ diverges n. In n. In In n n=2

A: Cauchy -Integral  Test series ∑n=1∞ an      converges if integral ∫1∞ f(x) dx finite and  diverge if…

Q: Suppose f(x, y) = /xy and R is the rectangle with vertices (0,0), (6,0), (6,4), (0,4). In each part,…

A:

Q: Let Chn), (tn) be fumction Sequence baundeel that Converges A. uni formely on on A that to h; t…

A: Given that &lt;hn&gt; and &lt;tn&gt;are the sequences of bounded function on A and are uniformly…

Q: 4. Let f : DCR → R be uniformly continuous and {rn} be a Cauchy sequence in D. Then show that…

A:

Q: Use the integral test to determine whether the following sums converge 1 2ninn n=2

A:

Q: 1 Determine the convergence of the integral dr. (2-r)

A: To determine convergence of the integral  ∫0512−x13dx

Q: Use the definition of the definite integral as the limit of a Riemann sum to evaluate L(7 - 4)° dr.

A:

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: Expand the function using Maclurin series expansion. f (x) 2 arctan y

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Q: Suppose f(x, y) = /xỹ and R is the rectangle with vertices (0,0), (6,0), (6,4), (0,4). In each part,…

A:

Q: Use the Raho Test to determine whether thc serics is convergen Or divergent oo(-3 n 0

A: Here, the general term of series  an is shown on board. Forstly, we calculate next term and…

Q: The integral S x3 converges to Select one: O True False

A:

Q: 4. Show that the series f(x) = E cos(nx) ni+ 2021 converges uniformly on R. Use this to show that f…

A: given sequence is                                                 we have to check convergence of…

Q: Let (hn), (tn) be sequences of bounded functions on A that converges uni formly Sh, tn) converges…

A:

Q: Let f : D ⊆ R → R be uniformly continuous and {xn} be a Cauchy sequence in D. Then show that {f(xn)}…

A: Uniformly Continuous If f: A  → R is uniformly continuous on A, then for given ε &gt; 0 there is δ…

Q: Exercise 5. Consider the sequence (fn)n defined by fn(x) = 1#nz: for r>0. -.... 3. Show that (fn)n…

A:

Q: b. Using right Riemann sums with 5 subintervals, approximate the net area bounded by the function…

A: We have to divide [-4,6] into 5 subintervals.  So each interval is of width= (6-(-4))/5= 2 So the…

Q: Use an appropriate Riemann sum to evaluate the limit

A: Solution:- We have to use an appropriate Riemann sum to evaluate the limit…

Q: 8. Use the integral test to determine if IM8 k² (k³ + 2)3/2 converges.

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Q: 4. Let f : DCR R be uniformly continuous and {r,} be a Cauchy sequence in D. Then show that {/(xn)}…

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Q: What is the Riemann sum of f(x) over [2,10]with n-4 partitions and mid-points for sample points?

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Q: Use the integral test to determine if -2 converges or diverges. Justify your en n n=1

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Q: 4. Use the definition of the definite integral as the limit of Ricmann sums to compute (Зл — 1) dr.…

A: We have to approximate the given integral using right endpoint formula.

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.