Let f.:A→R, j=1,2,... be a sequence of functions. When we say that the series f. converges uniformly to f, we mean that: j=1 n s →f uniformly, where s = Σ f, the nth partial sum of the series Σ n n O f→f uniformly. n n ○ f(x) →f (x) as j→ ∞ for each x € A. ∞ ∞ s(x) → f(x) for each x € A, where s_(x) = f(x), x € A is the nth partial sum of the series 8 WI f₁(x) .

Q: n+1 n+1

A: As per the instruction, we will be answering only the first one. For (1),     limn→∞…

Q: Let f be a positive, continuous, and decreasing function for x ≥ 1, such that a = f(n). If the…

A: Given Data: Let us consider the given data, ∑n=1∞1n2+1, twelve terms.

Q: 19. Give an example of divergent series E ak and E, bu such that the series E akbk converges.

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Q: Calculate the limit of the sequence an = Cos( 10πη3 – 3. 5n³+ n²

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Q: 1 1 E (ii) 1+ +9 n!' (iii) 1+4 +9+..., (iv) +) (i) +=) 3n n=0 .. ) n n=1

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Q: *? Let (sn) be a sequence that converges. (a) Show that if sn > a for all but finitely many n, then…

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Q: 4. Let ao 3 and an+1 = 2an + 12 an + 1 Find lim an n→∞ = a.

A: Since you have posted multiple questions, we will provide the solution only to the first question as…

Q: 1 (11) Σκ(F+2) k=1 (1.2) Σ k=1 4k (2k-1)(2k)(

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Q: .W: Which of the sequences {a,} con n+(-1)" 1. an %3D

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Q: 2. Prove that the following infinite series diverges. Σ(-1)*. k=1 (Hint: write down the first few…

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Q: Theorem 1.5. Consider the absolutely convergent series S(x) = ao + a1x + a2x2 + ...+ akx* + ·..,…

A: Given: The given series is Sx=a0+a1x+a2x2+.....+akxk+...

Q: on whernn 0o then det ermine this series E2 an is Convergent or Divergent n=2 -3 b. En=2 n2+n n2-10…

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Q: 3. Let for n € N. Is convergent? a₁ = 1, an+1 Σ n=1 = An sin an

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Q: n² – 5n 4. Use the Limit Comparison Test to determine whether the series converges n³ +n+ 1 n=1 or…

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Q: 3. Prove that the series 1 x2n n! 1 + x²n n=0 converges uniformly on R to a continuously…

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Q: Which of the following series conditionally converges? (−1)²+¹ 4 n (-1) ¹ √n 1. Σ n=1 1 II. Σ III. Σ…

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Q: IV. Determine whether the given series is convergent or divergent. Justify your answers. +oo +oo…

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Q: Which of the following sequences is convergent? Select one: O a. lim, v00 n² +1 3n /n15 n2 cos(1/n)…

A: b

Q: The sequence {Pn =}1 converges n-1 to p = 0 of order a = %3D O a. 3 O b. 1 О с. 2

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Q: 1. Determine whether each of the following sequences converges uniformly on [0,+∞). a. fn(x) =…

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Q: 1. Use any method to determine whether the following series converges or dive (a) E(-1)" n=1 (-1)"+1…

A: Note: Since you have asked multiple question, we will solve the first question for you. If you want…

Q: 4. Consider the sequence of functions {fn}=2 where fn: [a, b] R defined for n > 2 by if 0 < x <- fn…

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Q: \° over (0, 0) is x² + n?x Q.The sequence of functions 2x + 3n²x ) n=1 a. Not convergent.…

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Q: Example (21) prove the validity of the following series of functions with M. Specified regainst- A 3…

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Q: 6. Determine if the following series converge. If so, determine the limit. (a) E(1-(-1)*+2) i=1 (»)…

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Q: C. Determine the radius and the interval of convergence for the given power series: a. (x-2)3(x-2)5…

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Q: Consider the sequence S n²r(1 – nx) x € [0, 1/n] - fn(x) := 0. elsewhere . Show it converges…

A: Let gx=n2x1-nx Differentiating wrt x, we get  g'x=n2-2n3x Putting g'x=0, we get  n2-2n3x=0⇒x=12n