n + 3 2n + 1

Q: 1. Define when a sequence X = (x,) in R is said to converge to x E R. Then prove that the sequence X…

A: The given sequence is: xn=3n+12n+5

Q: B) Prove that the sequence is convergent, where az = v2, = v2, 2+an, an+1 = %3D n 2

A:

Q: Does the series 1+ 1/8+1/27+1/64+1/125+...(reciprocals of the perfect cubes) converge or diverge?…

A: we have to test whether given series is convergent or divergent

Q: Find all real numbers k so that the series (n – 3)* + (n – 3) (n³ – n³ – 1)'/3 n=4 converges.

A: Explanation of the answer is as follows

Q: Assume that the sequence converges and find its limit. a 1 = 7, an + 1 = 7an 7 -7 1 49

A:

Q: Prove {52 + 11/n} n=1 to infinity converges to 52 using the definition of a convergent sequence.

A: Prove {52 + 11/n} n=1 to infinity converges to 52 using the definition of a convergent sequence.

Q: Which of the sequences {an} converge, andwhich diverge? Find the limit of each convergent sequence.…

A: Consider the nth term for the infinite series.

Q: In(2n) 2n Determine the convergence of the sequence an If it converges, find lim am Paragraph BIU A…

A:

Q: (a) Provide an example of a sequence (an)neN such that a2m → 2 but (an)nen does not converge to 2.…

A: To find find the example of the given sequence that satisfy the above condition .

Q: 3 and Xn %3D 4-Xn-1 converges. %3D Prove that the sequence defined by x|

A:

Q: 1 Prove that if a sequence {a„} converges to a, then the sequence {[an]} converges to |a|. Find…

A: Suppose {an} converges to a. Prove that an converges to a. Also, check the converse is true.

Q: 1 The sequence converges to - 2 n² – 1 - Vn2 +n Select one: O True O False

A:

Q: Let an be an infinite sequence of real numbers. If an converges, then >-1 an converges. 'n=1

A: Consider the provided question, We have to check the given statement is true or false, Part first:

Q: 3. Suppose that the sequence (a,} is monotone, and that the sequence {a)} converges. Prove that the…

A: We have given sequence {an} that is monotonic  and sequence {an^2}  which is convergent. To prove…

Q: 1 The sequence converges to - 2 Un² - 1 - Vn² +n Select one: O True False

A:

Q: Q.1. Prove that if the series En=1an is convergent then the sequence an is convergent.

A: Let ∑n = 1∞an be an infinite series. We are given that the series is convergent. Let the series…

Q: (c) Consider the sequence (a,) with a, = (0,0, ...,0, 1, 1,0,0,...). %3! n-1 times i. Show that (a,)…

A:

Q: 1. Determine whether the following sequences converge or diverge. If the sequence diverges, state…

A:

Q: If a, converges and a, > 0 for every n, show that 4converges. What can you say about a for any…

A: Given: ∑n=1∞an converges and an>0 To prove: ∑n=1∞an2 converges. Proof: ∑n=1∞an2=∑n=1∞an2 Since…

Q: Determine whether the sequence converges or diverges. 3+5n² a. an n+n² b. - cos² n 2n

A: a.  Given an = 3 + 5n2n + n2 Here limn→∞an = limn→∞3 + 5n2n + n2              = limn→∞3n2 + 51n + 1…

Q: If {an} and {bn}1 are sequences of real numbers such that 0 < an < bn for all n 1and Σ An n=1…

A: A detailed solution is given in Steps 2 and 3.

Q: Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer…

A:

Q: (b) Define what it means for a sequence of real numbers (an) to converge to a real number t. Use the…

A:

Q: Assume that the sequence converges and find its limit. a 1 = 8, a n + 1 = VSan -8 64 8 1

A:

Q: 11. b) Determine whether the sequence an = 2- cos(n) is convergent or divergent.

A:

Q: Suppose that a, > 0 for all n and that an diverges. Suppose that b, is an arbitrary sequence of…

A: We have to solve this question with the help of comparison test:

Q: a. In a metric Space (X, d) if (xn) and (yn) are two sequences in X such that : X - x€ X and yn → y…

A: a) In a metric space (X,d) Let (xn) and (yn) be two sequences such that xn -&gt; x and yn -&gt; y…

Q: We say that a sequence of real numbers (xn)nɛN converges to the real number а, if Ve > 0,3N E N, Vn…

A: Given sequence is xn=-1n+2-n. Definition: A sequence of real numbers xnn∈N converges to the real…

Q: 12.7 Suppose that {x„}=1 is a sequence of real numbers that converges to xo and that all Xn and xo…

A:

Q: 3 A. Determine if the series =1212ria converges or diverges. If it is convergent find its value. В.…

A:

Q: 2 – 5n 11п-1 n=1

A: Use ratio test.

Q: Define a sequence (an) recursively by setting a1 = V2 and an+1 = v2+ an for n > 1. (a) Show that an…

A:

Q: (b) Give an example of divergent sequences {a,} and {b,} such that {a, + bn} converges.

A: To give an example The sequences {an} and {bn} are divergent sequence such that{an+bn} converges.

Q: 3 n° Cos n+ 5n J

A:

Q: 2. Consider the sequence (n+2)!. n=1 a. Show that the given sequence is monotone. b. Use the…

A:

Q: 2n+1 (2n-3 a.) = {(2²1 =3) "" } 116) (6.) = ( ³ + 05/(1+x)} 2n+5 11a) {a}={ O {c} = √√nº +3² +1 2n²…

A: Given: 11 a) an=2n-32n+52n+1, b) bn=3+cos3n+πn and c) cn=n6+3n2+12n2+3 To determine: Whether the…

Q: Let an = n 2 1 In(2" + 1) Determine if the sequence {an}n=1 is convergent or divergent 00

A:

Q: 1. Determine whether the sequence converges or diverges. If it converges, find the limit. 2n an %3D…

A:

Q: Determine whether the sequence converges or diverges an = In(n + 1) – In(n) %3D lim an %3D n- 00

A: Given sequence is an=lnn+1−lnn We have to find the limn→∞an Use property lnm−lnn=lnmn…

Q: 1. Suppose {Tn} is a sequence of real numbers that is (i) increasing (that is, In < In+1 for all n)…

A:

Q: { n2 100 3n+6 Jn=1

A: Explanation of the answer is as follows

Q: n· 3n 5n n=

A:

Q: Do the following: (a) Give an example of two divergent sequences (an) and (bn) for which (an + bn)…

A: Solution

Q: Prove if ( a n ) converges, then ( | a n | ) converges.

A:

Q: 1. Determine whether the sequence converges or diverges. If it converges, find the limit. 2n an 3n+1

A: Use limit ratio test to simplify and solve the problem easily.

Q: 2. Determine if the given sequence converges or diverges. If it converges, what is its limit? (e) an…

A:

Q: 3. a) Determine whether the sequence (an) = (V2n + 3 – V2n + 2) ,n e N converges or diverges? b)…

A: Explanation of the answer is as follows

Q: 1. Prove that the sequence diverges but it has a convergent subsequence: (-1)" 2+(-1)", (a) (b) X =…

A:

Q: 2. Determine whether the following sequence converges or diverges. If it converges, find its limit.…

A:

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.