6. True or False. If the statement is true, explain why using theorems/tests from class, and...
1. A series Can has the property that lim on = 0. Which of the following is true? (a) The series converges and has the sum 0. (b) The series is convergent but its sum is not necessarily 0. (c) The series is divergent. (d) There is not enough information to determine whether the series converges or diverges. 2. A sequence { $m} of partial sums of the series an has the property that lims Which of the following is...
For each statement: True or false? Explain? If the terms sn of a convergent sequence are all positive then lim sn is positive. If the sequence sn of positive terms is unbounded, then the sequence has a term greater than a million. If the sequence sn of positive terms is unbounded, then the sequence has an infinite number of terms greater than a million. If a sequence sn is convergent, then the terms sn tend to zero as n increases....
7. Determine whether the statement is true or false. If it is false, give an example that shows it is false. If it is true, prove it or refer to a theorem. (1) If {an} is divergent, then {an} is unbounded. (2) If {an} is bounded, then {an} is convergent. (3) If {an} converges and {bn} converges, then {an + bn} converges. (4) If {an) is convergent and {bn} is divergent, then {an + bn} is convergent. (5) If {an}...
Please let me know whether true or false If false, please give me the counter example! (a) If a seriesE1an converges, then lim,n-0 an = 0. m=1 (b) If f O(g), then f(x) < g(x) for all sufficiently large . R is any one-to-one differentiable function, then f-1 is (c) If f: R differentiable on R (d) The sequence a1, a2, a3, -.. defined by max{ sin 1, sin 2,-.- , sin n} an converges (e) If a power series...
1. A series has the property that lim an = 0. Which of the following is true? (a) The series converges and has the sum 0. (b) The series is convergent but its sum is not necessarily 0. (c) The series is divergent. (a) There is not enough information to determine whether the series converges or diverges. 1 n-00 2 2. A sequence {sn} of partial sums of the series an has the property that lim sn Which of the...
Determine whether each of the following is Always True, Sometimes True, or Always False. If the statement is Always True or Always False, provide a brief justification. If the statement is Sometimes True, provide an example of a series that makes it true and an example of a series that makes it false. In the following, {a_n}∞n=1 is a sequence and {s_n}∞n=1 refers to the corresponding sequence of partial sums. (a) If lim n→∞ s_n = 0, then lim n→∞...
1. Answer True or False, and give a brief justification for each answer: a) If lim 2 = 5 then the series i converges to 5. b) If = 5 then lime = 5. c) If S. and lim.- S.-5, then 10 -5. d) The series 5-5+5-5+... is divergent. e) If = 0 = 5 and the = 5, then 20 - 5 f) The Divergence Test can be used to prove a series is convergent.
DIRECTIONS: Show all of your work and write your answer in the space provided. MODIFIED TRUE/FALSE: If the statment is true, write true in the blank. If it is false, replace the underlined word(s) with the word(s) that will make the statement true. 1. A series that tends toward a single number is called a divergent series. 2. A series is the product of the terms in a sequence. 3. A(n) alternating geometric sequence switches between positive and negative values....
Please only answer questions a, d, and f. Thank you. 1. True/False Explain. If true, provide a brief explanation and if false, provide a counterexample. Choose 3 to answer, if more than 3 are completed I will pick the most convenient 3. Given a sequence {an} with linn→alanF1, it follows that linnn→aA,-1. b. A series whose terms converge to 0 always converges. c. A sequence an converges if for some M< oo, an 2 M and an+1 >an for all...
for these 2 theorems, pick on hypothesis, remove it, and provide a counterexample showing that the new statement is false. (d) Thm The Weierstrass Uniform Convergence Criterion): The sequence of functions converges uniformly to some f: D R iff the sequence In is uniformly Cauchy ,, : D (c) Thm (Differentiation of Power Series): If a power series converges on (-r,r) then it has all derivatives there and those derivatives may be found by differentiating the power series term-by- term....