Suppose ak > 0 and there exists M >0 so that E ak < M for all n E N. Prove that ak converges. k=n 0=:

Q: For each natural number n, let the function fn: R → R be bounded. Suppose that the sequence…

A:

Q: Which of the following statements is true? O E )" converges to - 2-1 7 12 O E O"' diverges n-1 O E…

A: The geometric series 1+ar+ar2+ar3+....…

Q: p for which In (n) Comvenges

A: Given∑n=1∞ln(n)np

Q: It is possible to find a value for a so that the series 00 1+ n° Σ na+1 n=1 is convergent. Şelect…

A:

Q: For what values of x does the series converge conditiona 10x" Σ n= 1

A: for a series to be conditional convergent ∑an has to be convergent and ∑|an| should be divergent.

Q: 1. Use an appropriate way to determine whether the following is convergent or divergent. k 'Ink i. Σ…

A:

Q: In(n + 9) State whether the sequence ɑn = converges and, if it does, find the limit. n а) converges…

A: Given data is : an=lnn+9n

Q: Suppose that u is a series of positive terms and that g is a continuous function that is…

A: True

Q: The infinite sum Ea(n) A) S = a(n) has a finite limit B) none of these %3D converges if and only if

A: ∑n=1∞a(n) converges

Q: Write the first five terms of p_j= j/(j+1) and then prove p_j converges to 1. Be sure to explicitly…

A: We have to solve an inequality for apsilon and j.

Q: Given: a, = In(n + 1) – In(n) for n E N || Determine if the following is convergent or divergent.…

A: Given  an = lnn+1 - lnn for n∈ℕ.

Q: Solve the problem. If Tan is a convergent series of nonnegative terms, what can be said about nk an…

A: Introduction: The p-series is a significant infinite series. It is widely used when we compare a…

Q: Given: an = In(n+ 1) – In(n) for n E N | Determine if the following is convergent or divergent. (b)…

A:

Q: Prove: if 0 is less than or equal to r which is less than or equal to 1, then (r^n) converges.…

A:

Q: Suppose that the power series ∑∞ n=0 an(x − 2)^n converges for x = 5. At which of the following…

A:

Q: The sequence a, converges to the limit A) e B) ve C) Ve D) e2 E) F) e2

A: Given : The sequence     To find the convergence of the given sequence.

Q: For each nN, consider x,, H (a) (b) 71 X with x E R. Define the sequence {gn} by 8n(x) = f(x) = 1…

A:

Q: Prove the following: If (xn) converges to x, if x 0, and if rn 0 for all n 2 1, then there Prove…

A: As per our guidelines we are supposed to answer only first question please repost other parts in the…

Q: Give an example of each of the following, or state that such arequest is impossible by referencing…

A: Let xn=n and yn=-n Both xn and yn are unbounded and therefore diverge However, xn+ yn=n-n=0 Hence…

Q: Given: an = In(n + 1) – In(n) for n e N Determine if the following is convergent or divergent. (a)…

A:

Q: Which of the following is TRUE for (-1)*+1 ? 00 Lütfen birini seçin: a.Diverges conditionally…

A: We have given series ∑n=1∞(-1)n+1n2en This is an Alternating series, We know the alternating series…

Q: Given: a, = In(n + 1) – In(n) for n e N Determine if the following is convergent or divergent. (a)…

A: Given that an =ln(n+1) - ln(n) We have to find whether the sequence ann=1∞  is convergent or not. A…

Q: Let k be a positive integer. Prove that the series of functions 00 En*(x- In(x + 1))*™ n=1 converges…

A:

Q: For which values of a the power series 00 n=1 converges absolutely and for which values of a it must…

A:

Q: Suppose that > an(x+9)" converges at a = 13. At which of the following values of x must the series…

A: Consider the given information.

Q: (k!)* (2k)! k=1

A:

Q: If lim M→0 J1 rM dx converges, then which of the xP Following must be true? elect one: o a. Σ…

A:

Q: For n e N let fn: [0, 1] → R be the function fn(x) = e²-n. Decide whether the sequence (fn)converges…

A:

Q: k en(k) 大=Z

A:

Q: If (am) is increasing and en (an) is bounded by M converges to a, {|a1\, |a|}. max

A: Let (an) is increasing sequence and converges to a,

Q: Use sequence

A: converges.

Q: Given that " In (1) converges, does [.. converge or diverge? In(n) 1 (²+5)² A. Diverges B. Cannot be…

A: Let ∑pn and ∑qn are two series such that pn≤qn. If ∑qn is convergent, then ∑pn is convergent.

Q: Find the interval of convergence of the series E*. k=1

A:

Q: Set an where r E R is a real number. Prove that ɑn converges to 0 if (-1, 1), cônverges tó 1 if r =…

A:

Q: Use the limit comparison test to determine whether the series E=1 * (k+1)(k+2)(k+5) converges. A…

A: The given series: ∑k=1∞kkk+3k+1k+2k+5 Limit comparison test:  Consider two series ∑nan and ∑nbn. If…

Q: (e)M 00 4n+1. n=1

A:

Q: The following sequence (7532h n! an = e O diverges converges to e converges to 0 converges to 1

A: Given problem:-

Q: If E a, is a convergent series of positive terms, prove that An is convergent. 1+ an

A:

Q: The series (k+1)* k=1_p*k! IS a) Converges absolutely b) Conditionally convergent c) Diverges d)…

A: The given series  ∑k=1∞k+1kpkk! is ; a) Converges absolutely b) Conditionally convergent c) Diverges…

Q: The sequence {C+1 In=2)"} N=1 converges to e² O None diverges converges to le converges to e³

A: The given sequence 3n+23n+1nn=1∞ The sequence can be written as follows:…

Q: Write the first five terms of p_j= 1/j and then prove p_j converges to 0. explicitly write the…

A: Concept:

Q: Show that: For any f E C(R/Z; C), the If (n)|² is absolutely convergent, and n=-00

A:

Q: Suppose E-1 an and E=1 bn are series converges absolutely. Prove or give counterexamples for the…

A:

Q: State the Weierstrass's M-test, for the series of functions. Hence show that each of the following…

A:

Q: Justify whether the following sequence {an} and the series an converges. If yes, find its limit. (a)…

A: