Let b>0. Is the series of function below uniformly convergent on (b,b+1) ? Give complete proof. > [(x – b) · In(x – b)]" n=1

Q: n Use the integral test to decide whether the series - n²+1 converges or diverges. We determine…

A:

Q: Σ n=1

A: Let the given series be: ∑n=1∞an where, an=3n(n+2)!n5

Q: 28. (Continuation of Exercise 27.) If E=1an converges, and if 1 > a, > 0 for all n, show that E=1 In…

A:

Q: 00 (-1)"+1 Is En=1 convergent absolutely, non-absolutely, or diverges. n+8

A: Given series is ∑n=1∞-1n+1n+8 Absolute convergent test: If ∑an converges, then ∑an converges. Let…

Q: Show that f"(0) = 0. Use induction and the ideas from the above to show that f(k)(0) = 0 for every k…

A: Given that fx=∑n=0∞anxn in the interval -R,R. Then R is the radius of convergence of the function f.…

Q: (x + n)² Show that h„(x) converges poimtwise to f(x) = 1, V x e [1, 00). %3| x² + n²

A:

Q: n is convergent. Be sure to check n² 2 Use the integral test to determine if n=2 that the conditions…

A:

Q: a. Show that if two power series E-oa„x" and E-o b,x" are convergent and equal for all values of x…

A:

Q: (-1)"n" sin" (4) 2n n=1 CS Samed with Camacarner

A:

Q: Prove that the sequence xn has a convergent subsequence. 3n4 sin n – 4n cos n2 + 2 a) Xn 5n4 – 8n –…

A: We have to prove that xn has a convergent subsequence in each of the given cases.

Q: (vi) 5. Do the following sequences converge as n → 0? Prove or disprove. 1/2 (i) 1/n en 1,6 /…

A:

Q: 6.3 Represent the function f(x) =,, using a power series and find the interval of convergence. 2 – x

A:

Q: +o0 ΣΗ 1 -3 -1-n n=1

A:

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: Solve Σ 0 to ∞ ((n! • x^n)/ n^n) for x = e and x = -e and explain whether each point converges or…

A: Solve for x = e and check whether it converges or diverges as follows.

Q: 00 Let r be a constant. Suppose ' converges to -1, then the sum of )r' is i-1 i-0

A:

Q: Determine if the given is convergent or divergent. (x²+1)² n Zn=1 (n²+1)²

A:

Q: 1. Find a power series representation for the function f(x) = 5x2- Determine the interval of…

A:

Q: E (1- nsin-) converges n = 1

A: Series convergence test problem.....

Q: 24.6 Let fn(x) = (x − 1)² for x = [0, 1]. (a) Does the sequence (fn) converge pointwise on the set…

A:

Q: 1 Use appropriate tests to determine if E(-1)"+1, converges. n* log(n) + n! n=1

A: use appropriate test to determine whether the series ∑n=1∞-1n+11n4logn+n! converges according to…

Q: If 2 Cnx" is convergent when x=2 and divergent when x=-3, then 2 Cn is n=0 n =0 convergent. O…

A: Consider the given information: The series ∑n=0∞Cnxn is convergent when x=2 and divergent when x=-3.…

Q: show : ΣΣ - a"d, (x - a)" | n=0 m=0 is convergent,

A:

Q: (k + 1)/2 (k5 + k3 – 1)/3 1. Show that converges. k=1

A:

Q: Show that the sequence of functions {f.} where f,(x)=nxe is non-uniformly convergent on [0,1]. (by…

A: Test for uniform convergence of sequences: Let fn be a sequence of functions such that it converges…