1. Use the limit theorems to prove that the following sequences converge. 15n² 6 sin (2n + 1). ; (2+2), n an 5n2 1 19 bn Cn=n= n; dn gn (n+8)!*

Q: Prove that the sequence {sin n + cos n is bounded. =1

A: The given sequence is We have to prove that the sequence is bounded.

Q: Consider the sequence In+1 n = 0, 1, 2,... (a) Show that this sequence is generated by Newton's…

A:

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

A:

Q: 1 1 E (ii) 1+ +9 n!' (iii) 1+4 +9+..., (iv) +) (i) +=) 3n n=0 .. ) n n=1

A:

Q: Jse the limit theorems to prove that the following sequences converge. 15n2 – 6 sin(2n + 1). bn n +1…

A: Note: Our guidelines we are supposed to answer only three subpart . Kindly repost other subpart as…

Q: n cos(n) V1+4n2 (an) at n→∞o or prove that the limit does not exist: (b) In(1+2n²) - 2 ln(1 + n)

A: 7) a) an=ncos(pi n) /sqrt(1+4n2)b) an)-2ln(1+n)

Q: Match the sequence with it's limit as n- 00 2n+1 A. 00 3n2+2 В. О n! C. 2 D. 2 (n-1)! v 2n+2 3n+1 (-…

A:

Q: 2.2.1. Prove that each of the following sequences converges to zero. write a) xn =…

A: We have to show that the following sequences  converge to zero.

Q: 2) Suppose that f(x) = 5x² over the interval [1, 2]. Let Ax = 1/n. Let x₁=1+iAx. Find the limit of…

A: Given: fx=5-x2.Formula to be used: ∑i=1ni=nn+12 ; ∑i=1ni2=nn+12n+16

Q: In(1+2e") d) an n

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

Q: Prove by contradiction the scquence {(-1)"+} is divergent.

A:

Q: 3. Find the limits of the following sequences as n → ∞. List the theorems you use. an = e¬" sin n •…

A: The sine function is defined on all real values and the range is [-1,1]. The value of the sine…

Q: 2. Use limits to determine if the the following sequences converge or diverge. If the sequence…

A: As per our company guidelines, we are supposed to answer only the first 3 sub-parts. Kindly repost…

Q: 6. Consider the sequence of functions n²a* – sin(nx) fn(x) = x E (0, 00) n2x (i) Show that fn…

A:

Q: 1. Recall the definition of the limit of a sequence of real numbers (xn) = (x1, X2, X3, . . .) "(Tn)…

A:

Q: 3n 14 Determine the rate of convergence of the sequence an sin as n → 0. A. O(1/n^2) B. O(1/n^3) C.…

A:

Q: 2. Determine the convergence and divergence of the following Sequences: 3" 6" b. (1 -)") c. flan a.…

A: Note : Our guidelines we are supposed to answer only first three subpart . Kindly repost other…

Q: 1 Given that the series 2 (1n² + 1)’ 7 converges where P > which of the following is true *-5 O a.…

A: Given series ∑n=5∞6nn2+1p converges where p>12 We have to check which of the given option are…

Q: For the sequence an = n cos n n² +1

A: given a sequence

Q: Consider the following six sequences: Cn = n², dn 6n + 4 7n 3' Wn = n Un - = (-²) * · (27). —2)", Zn…

A: The objective of the question is to analyze the given sequences in terms of monotonicity,…

Q: Consider the sequence S n²r(1 – nx) x € [0, 1/n] - fn(x) := 0. elsewhere . Show it converges…

A: Let gx=n2x1-nx Differentiating wrt x, we get  g'x=n2-2n3x Putting g'x=0, we get  n2-2n3x=0⇒x=12n

Q: For each of (a)-(d) give a sequence (rn) with the stated prop- erties. In each case, the sequence…

A: Example and illustrations.

Q: Let f(x)= (-1)* x« (2k)! ,6k Find [f (x)dx. Write in summation form. k =0

A: Here is the solution of the given problem.

Q: 4. Determine whether the following sequences converge. Rigorously justify your answers. You may use…

A:

Q: 2. If L, and S, is its n-th partial sum, find: 3 (a) lim cos(an) (b) lim S

A:

Q: 00 Find the sum of the series> by finding the limit of its partial sums.

A: