100 For the infinite series S = Σ1 an, if an = (1 + )", which statement is true? S is divergent, since when OS is convergent, since when S is divergent, since when S is convergent, since when → ∞o, lim an = e. n→ ∞o, lim an = 0. → ∞, lim an = 0. n→ ∞, lim an = e.

Q: Let {an} be a positive sequence such that lim Van 1 Determine which of the following 3 n00 series…

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Q: Suppose that a, converges, but not absolutely. Then (a) both series (an] ±an) = ∞o, and (b) liman =…

A: Suppose that ∑an converges, but not absolutely. Then a) both series ∑an±an=∞, and b) limn→∞an=0. We…

Q: 14. Three of the following series are convergent. Circle the one that diverges? а. n=1 3n b. 2 '2"…

A: we have to circle the one that diverges

Q: Let {an} be such that an 0 and let lim an+1| JanT = 0.5. Which of the following statements is always…

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Q: co s(n.n) 4- The series En=1 (n+1)2 A- Converges, B- Diverges, C- None

A: ∑n=1∞cosnπn+12 Here, an=cosnπn+12=-1nn+12 Now comparing with an=-1nbn, we get bn=1n+12 Now,…

Q: 2" + 3" (a) Show that the series converges. 4" + 5" k2 k=1 n=1

A: Note: According to bartleby we have to solve the first question please upload the question…

Q: Let x0, x1,x2,... be the sequence such that xo = 1 and for n ≥ 0, Xn+1 = ln(en — Xn) (as usual, the…

A: Let x0, x1, x2, ...... be the sequence such that x0=1. For n≥0, xn+1=lnexn-xn. We need to show that…

Q: . For n = 0, 1, 2, 3,..., let an = (a) Find lim sup(an)¹/", lim inf(an)¹/", lim sup|ªn+¹| and lim…

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Q: If |r| < 1, the series a +ar+ar² +...+arn−1 t... converges to

A: Series convergence.

Q: 3. Prove the statement: If the series > an is convergent, then a, →0 as n→ 0. Is it also true that…

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Q: For the infinite series series n=1 n=1 1 2n it is clear that lim 1 is convergent? Why or why not? 2n…

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Q: 4) Let Eanbe given series and let Eb, be series in which the terms are the same order as in Ean…

A: Given- Let ∑an be given series and let ∑bn be the series in which the terms are the same order as in…

Q: ‘?

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Q: Which of the following infinite series converge? n(n +2) i) En=1 (n +3)³ * sin (ln n) ii) n=1 n…

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Q: Which of the following infinite series converge? 3n(n2+2) En=1" (n +2)3 1 ) En=2 n(Ln n)2 tan-(cos…

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Q: 2 With the aid of series, prove that if f(z) = COS Z Z-π/2 −1 when z = π/2, when z = π/2 then f is…

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Q: 8. If lim a,=0, then the infinite series 2 a, must converge k=1 Select one: O True ion O False…

A: Given query is to find statement is true or false.

Q: A geometric series E ar" -1 is convergent if n=1 O A. None of these O B. Ir| = 1 O C Ir| >1 O D. Ir|…

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Q: For what values of x does the series, (-1)"(3 – x)" n4n n=1 a) absolutely, b) conditionally?

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Q: 8. Consider the geometric series En=1 the third term az = 12. Which of the following statements…

A: Geometric series.

Q: 00 Use the equation >x" for |x| < 1 to expand the function 2 in a power series with center c = 0.…

A: Interval of convergence is (-1, 1)  because at x=1,-1 series is not convergence because it is not…

Q: Σ'an where an n=1 = (−1)n+5 ( n² n? + 6 n? +5,

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Q: an Suppose 2 a, and 2 bn are series with positive terms. If lim = c, where c is and value of C is…

A: We have to complete the statement

Q: For the infinite series 1 n!' antl limn 00 an Select one: a. and hence converges according to the…

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Q: The series ∞ 1 n[In(√n)]² n=2 n is convergent O is divergent O it is not possible to determine if it…

A: To find the convergency of the given series

Q: 6. Which of the following is false? a. If the series Σn an converges, then lim a, = 0. 818 b. If lim…

A: Only Option C is False.

Q: Let 2 ak converge with the positive Ak Converge with the positive term- In this case examine. the…

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Q: 10. Which of the following is/are correct? an is convergent. na n=1 I. For a = 2 the series II. For…

A: Option B is correct.