2n2 + 3 Let Sn ak. If Sn for each n > 1, which of the following statements is n2 +1 k=1 correct? (a) lim an (b) lim an =2 n00 (c) 2 an is convergent and its sum is 3 n=1 (d) an is divergent. n=1

Q: 2n - 4n² –5 1. Use the limit theorems (or homework from 4.1) to show that lim converges. Hint: Do…

A: According to experts guidelines of bartleby i have to solve only first question so repost for…

Q: an a) Converges n2 b) lim an =0 > cans converges dd) > cos (an) Converges n=1 e) Cos can) diverges

A:

Q: 11. If {an} and {b„} are sequences of real numbers, lim,0 an = 5, lim, bn = 8, and fn(x) = a, + bnx,…

A:

Q: 3n + 4 A) B) none of these 4 10n 3n +4 lim 4 10n #0 has NO HOPE of convergence since n00

A:

Q: A sequence (xn)∞ {n=1} is said to oscillate if lim inf xn < lim sup xn.  Prove or disprove the…

A: We can solve this question using definition of converges diverges and oscillating of sequences

Q: 1 1 Зп — 1 3n + 2 n=1

A:

Q: Let a, = 2n+1– 2". Then, lim an %3D n-00 None of the other options the above co and the sequence…

A:

Q: 4n – 9 5n + 4 S - Given that sequence {an} = { (а) Show that the sequence is monotonically…

A:

Q: Consider an = V4n2 + 3n – 2n for n 2 1. - (a) Determine if {an}, n=1 is convergent or divergent (b)…

A: Here, an=4n2+3n-2n, n≥1     =4n2+3n-2n4n2+3n+2n4n2+3n+2n     =4n2+3n-4n24n2+3n+2n     =3n4n2+3n+2n

Q: (2k + 6)! Consider the series > ar | k-1((k + 4)!)2" k-1 a) What is ak+1? ak-1 = ak+l? b) What is ak…

A:

Q: an+1 (a) lim = 9 n- 00 an O absolutely convergent O conditionally convergent divergent cannot be…

A:

Q: 6n + 2 6n+2 Let > and an 5" 5" n=1 an+1 a) Find lim n-0 an 6n+2 b) Determine whether 2 is convergent…

A: Convergence of series.

Q: 2. Let {an} be a decreasing sequence of positive numbers with limit 0. I define a new sequence {Tn}…

A: Given: ann=1∞ be decreasing sequence of positive numbers with limits 0Let x1= a1 ;   ∀ n∈ℤ+ ,…

Q: 8 (iv) [(-1) n=1 n :)" sin n n+1 [ Select] [Select] is conditionally convergent is divergent. is…

A:

Q: 8. 4" + 1 7" + 1 n = 1 4n + 1 7n + 1 lim = L > 0 converges diverges

A: Follow next step

Q: 2n Let an = 3n +5 Determine whether {an} is convergent. Determine whether an is convergent.

A: To determine whether the given sequence and series is convergent or not.

Q: -n 1 1+ 5n n=1 L = lim Vlan| (Enter 'inf' for .) E (1 + )" is: En=4 -n is: 5n OA. convergent B.…

A: According to question,Given series,∑n =1∞1+15n-n

Q: (3k + 8)! Consider the series> ar k3k k-1 k-1 a) What is ak+1? ak+1 = aktl ? b) What is ak ak-1/ak =…

A: The given series,     ∑k=1∞ak=∑k=1∞3k+8!k3k

Q: Let a, = vn +1– yn. Then, lim a, = None of the other options -00 and the sequence (a,,} is divergent…

A:

Q: 1.1 Let {fn(x)} = { ,x ER and r > 0. %3D + xn 1.1.1 Find the pointwise limit of the sequence {fn(x)}…

A:

Q: Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is…

A: Here, the given series is: ∑n=1∞7n2(2n+1)! Let, an=7n2(2n+1)!; n≥1

Q: The integral tests says that if an=f(n), then the series 2 an is convergent if and only n =1 if the…

A: for integral test three conditions must be satisfied for the function i.e.  continuous, positive and…

Q: Suppose xo = 2√3, yo = 3, 2xn-14-1 Xn-1+Yn-1 and Prove that (a) xnx and yn ↑y as n→ co for some x, y…

A: The statement of monotone convergence theorem is if a sequence is monotonically increasing and…

Q: Let an = 2"+1– 2". Then, lim an = %3D n-00 None of the other options O the above co and the sequence…

A:

Q: a) They are all convergent b) 1 is convergent but II and III are divergent © II and III are…

A:

Q: 2n2+3n c) Identify whether the series En=1 es is converges or diverges by the Limit 5+n5

A:

Q: 1. The sequence {V(n + 4)(n + 6) – n} is A) Convergent to 0. B) Divergent. C) limn-o V(n + 4)(n + 6)…

A: the sequence given is (n+4)(n+6)-n

Q: 2n+1 3. Test the convergence of E=1 by limit comparison test. n%3D1

A:

Q: 3 If Xn 4 and ly,l0. Select one: True O False

A: Given, xn → 4  and  yn≤31n2n for all values of n To determine whether ∑ xnynp  is absolutely…

Q: the series %3D an V52n+1 n=1 n=1 is divergent since lim an= %3D n- +o 16 This option O This option…

A: 5^lim(2n+1)/(n)=5^2=25.

Q: n!n! is absolutely convergent using the Ratio Test. (2n)! 2. Prove n=1

A:

Q: (b) Let X and explain whether the sequence X = (Xn) converges or diverges. (xn) be defined by xn =…

A: We have to solve 4.b

Q: n = 1 Vn lim n-00 converges O diverges

A:

Q: Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is…

A:

Q: nx Let {fn(2)} = {1 for x ≥ 0. 1+nx . (a) Find lim fn(x) = f(x). n-x (b) Show that if t > 0, the…

A:

Q: 00 Consider the series (-2)" Find the limit an+1 lim (2n-1)! an n=1 n- 00

A:

Q: -1 6) Show that tan N. converges/di verges. N2 N-1

A: Solution of question as follows

Q: 1.1 Let {fn(#}} = {1 ng „I ER and r > 0. 1+ nx 1.1.1 Find the pointwise limit of the sequence {fn…

A:

Q: (4) Suppose xo = 2√√3, yo = 3, 2xn-19n-1 Xn-1+Yn-1 and Prove that (a) xnx and yn ↑y as n→ ∞o for…

A: Hints:  It is given that, x0=23, y0=3 and xn=2xn-1yn-1xn-1+yn-1 and yn=xnyn-1 for all n∈ℕ.…

Q: 1 1 = 0. Vn – 1+ /n converges because lim Vn – 1+ n n=1 True False

A: Since you have asked multiple questions in a single request we would be answering only the first…

Q: 9n Let an Зп + 1 (a) Determine whether {a,} is convergent. convergent O divergent (b) Determine…

A:

Q: 0O 5n -2n n + 1 n = 2 Identify an 5n -2n n+1 Evaluate the following limit. lim VTanl n- 00 32 Since…

A:

Q: In +4 7n+4 Let 2 5" and an 5" n=1 an+1 a) Find lim an n + 00 7n+4 is n =1 5" b) Determine whether 2…

A:

Q: 1 25 +1 125 - O convergent divergent If it is convergent, find its sum. (If an answer does not…

A: Explanation of the solution is given below...

Q: 13. Which of the following statements is/are false? I. En1 ar"-1 is divergent if |r| > 1 convergent…

A:

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

A:

Q: (-1)"(7n + 1) Does En=1 converge or diverge? (3n + 1)

A: Alternating series test: If infinite series is If any of these two conditions does not hold, then…

Q: 6) If an converges, then n=1 a) E converges. n=1 b) lim an = 0. n00 00 c) E(an)² converges. n=1 d) E…

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

Q: Let a1 = 1 and an+1 = [1-1/(n+1)^2]an for all n ≥ 1. 1) Show that the sequence {an}n≥1 converges.…

A: