2. For n E N, let fn(2) - converges uniformly on [0, 1]? nx +sin(nx?) n 1 (a) Show that the sequence {f}₁ converges uniformly on [0, 1]. (b) Does 8]T Σ n=1 fn

Q: 11. If {an} and {b„} are sequences of real numbers, lim,0 an = 5, lim, bn = 8, and fn(x) = a, + bnx,…

A:

Q: Let f be differentiable on R with a = sup{|f'(x)| : x ≤ R} < 1. = f(so), $2 = f($1), etc. Prove (Sn)…

A:

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

A:

Q: 00 n-1/2 6 + cos (n) A) diverges B) converges

A: Given series is: ∑n=1∞n-126+cos2n

Q: For each nЄ Z+ let fn : R → R be the function пх fn(2) n|x| +1° (a) Prove that (fn) converges…

A: Given the sequence of functions  defined by .(a)Consider the limit,This is true for every 0≠x∈R.…

Q: Consider the sequence of functions {fn}=2 where fn: [a, b] → R defined for n 2 2 by if 0<x s -n² (x…

A: Definition: Suppose fn is a sequence of functions fn:S→ℝ and f:S→ℝ. Then fn→f pointwise on S if…

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: 1. (a) Let an → 2. Prove from first principles (i.e. give a direct e-N proof) that a → 4. (b) Let a?…

A:

Q: Find the limit (as n → ∞) of the sequence an = In(2n² + 5) – In(5n² + 2), n > 1. %3D

A:

Q: 3. Let for n € N. Is convergent? a₁ = 1, an+1 Σ n=1 = An sin an

A:

Q: 3. Prove that the series 1 x2n n! 1 + x²n n=0 converges uniformly on R to a continuously…

A: We'll follow these steps:Show that the series converges uniformly on R.Prove that the limit function…

Q: For each n ≥ 1, let fn : R → R, fn(x) =nx/(n + x^2) Show that the sequence of functions {fn}n≥1…

A:

Q: Match each of the following with the correct statement. C stands for Convergent, D stands for…

A: Concept used: Divergence: When the values goes to -∞ or goes to ∞ that is it does not approach one…

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

A:

Q: 2n Let ɑn en O {an} is divergent. O {an} is convergent. 2 {an} converges to e O {an} converges to 0.

A: A detailed solution is given below.

Q: For eksample Xn = n² (n+1)! investigate whether the serles Σ 8 n=1 Xn convergent Or divergent. Prove…

A: We will use ratio test to check whether given series is convergent or not.

Q: 4. Consider the sequence of functions {fn}=2 where fn: [a, b] R defined for n > 2 by if 0 < x <- fn…

A:

Q: Suppose that (sn)neN and (tn)neN are sequences and assume that (tn) converges. (a) Suppose that…

A:

Q: Let (Jn)neN be a sequence of functions f: R → R satisfying |fn+1(c) – fn(c)| 2 and neN , and ae(0,…

A:

Q: Let ƒ : [−1,1] → R be a continuous function. Prove that the sequence (n) defined inductively by Xn 1…

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: 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: 1 Prove: Ex-1 n=1 (n+1)(log(n+1)), converges for p > 1 and diverges for p ≤ 1.

A:

Q: Use the definition of convergence to prove lim COS n 72-00 n²n+1 = 0 || -

A:

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: Suppose {fn}n=1 and {gn}n=1 converges uniformly on E where {fn}n=1 and {gn}n=1 are sequences of…

A:

Q: finding if this converges or diverges

A: