(n² + 2n) (n³ – 5) Use the definition of convergence to prove: lim

Q: 1 a) lim n+00 V 0. 8 ) lim b) noo n+3 0. 5n2 – 3 (c) lim n0o 9n2 +4

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Q: Let a, b E R with 0 < a < b and let n = Van + bn. Prove that (n) converges and determine lim (x₂).

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Q: 2. Use an idea similar to the Squeeze Theorem (the Push Theorem?) to investigate the conver-…

A: The given sequence is, nnn!. The squeeze theorem states that if fx≤g(x)≤h(x) for all the numbers and…

Q: Prove that the family of all polynomials of degree < N with co- efficients in (-1, 1] is uniformly…

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Q: Given a sequence x₁, x2,... of real numbers, show that cos(x₁) + + cos(xn) exists if and only if…

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Q: Use the definition of convergence to prove lim {n→∞} (cos n)/(n2-n+1) = 0.

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Q: 2. Prove that the sequence {n} defined by xn := (-1)n 3n+2 is divergent. Use an argument similar to…

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Q: Let an be recursively defined by ao = 2, an+1 = √5an for n ≥ 0. Either show that an converges and…

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Q: Consider the sequence In(1/n) An V2n first five terms of an , and find lim„→c∞ An - If the sequence…

A: Here we have ,  an=ln1/n2n Here we have to find first five terms,

Q: b) Suppose (xn) does not converge to 0. Then there exists e > 0 such that all except for finitely…

A: Let us suppose the following sequence xn=1,0,1,0,1,0,..... We have to see that the odd terms are 1…

Q: 4n+1 Σ Z 3n – 2 3п — X=1

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Q: Given ſtan xdx = limn→∞ Σ-1 (tan 7)) (7) identify if it is a right k=1 Riemann sum. O True O False

A: The right Riemann sum of integral

Q: Suppose that to ER and n = Theorem to prove that lim n = 1. n→∞ 1+xn-1 2 for n E N. Use the Monotone…

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Q: Let (an) be a sequence of real numbers that diverges to +∞, and let (bn) be bounded. Prove that lim…

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Q: For eksample Xn = n² (n+1)! investigate whether the serles Σ 8 n=1 Xn convergent Or divergent. Prove…

A: We will use ratio test to check whether given series is convergent or not.

Q: VO 0 4 2 C n - 1 and use the Binomial Theorem to Prove that limon = 1. (Hint: Define an show that…

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Q: 3. Prove that for any bounded sequences (n) and yn we have lim sup(n + yn) ≤ lim sup(x) + lim…

A: We have to prove that limsupxn+yn≤limsupxn+limsupyn. Let xn and yn are sequence of R. Let limsupxn=x…

Q: (iii) 2 an must converge if lim an 0. n =0 bn (iv) 2 bn must diverge if b, > 0 and lim = 4. n->0…

A: The solution are next step is

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.