anjali kattaru

2a^2 + 2b^2 + c^2 + 4ab + 3ac + 3bc

An online bookseller charges $3 per order plus $1 per book for shipping. John places an order for four books that have the same price. The total cost of his order is $30. What is the price in dollars of each book?

a company has $1500 in its budget for paper this year. The regular price of paper is $32 per box with a 10% discount on bulk orders. If the company spends at least $1400 on paper the shipping is free. What is the least possible number of boxes the company can buy with the discount to get free shipping?

Prove that a convergent sequence {x_k} in R^n is bounded.

Suppose that A is an open set in R, and that A ∩ B = Ø for some set B. Prove that A ∩ (closure of B) = Ø.

∂(A ∩ B) ⊆ ∂A ∪ ∂B

(i) Let S = {(x, y) ∈ R^2 : 0 < x < 1}. Prove that S is an open set.

(ii) Let S = {(x, y) ∈ R^2 : 0 < x <= 1}. Prove that S is not an open set.

(i) Let S = {(x, y) ∈ R^2 : 0 < x < 1}. Prove that S is an open set.

(ii) Let S = {(x, y) ∈ R^2 : 0 < x <= 1}. Prove that S is not an open set.

Let A ⊂ R^n and define N_ε = {x ∈ R^n : dist(x, A) <= ε} where ε > 0.

(i) prove that N_ε is closed.

(ii)Prove that A is closed if and only if A = ∩{N_ε : ε > 0}.

Let A ⊂ R^n and x ∈ R^n. Define dist(x, A) = inf {dist(x, y): y ∈ A} and for each ε > 0 let D(A, ε) = {x : dist(x, A) < ε}. Prove that D(A, ε) is open.

Let K = {(x, y) ∈ R^2: xy >= 0}. Prove that K is a closed subset of R^2

Let A ⊂ R^n be an open and B ⊂ R^n. Defi…ne A + B = {a+b : a ∈ A and b ∈ B}. Prove that A + B is open.

Prove that the set of rational numbers Q is neither open nor closed.

Prove that R \ N and R \ Z are open sets.

(a) Prove that if ∑(k = 0 to ∞) (a_k) converges to L, then ∑(k = 1 to ∞) (a_k) is Abel summable to L

(b) Find the Abel sum of ∑(k = 1 to ∞) ((-1)^k).

Define f(x) = ∑(k = 1 to ∞) [1/(x + k^2)] for x belong to [0,1]

(a) Prove that the series converges uniformly on [0, 1].

(b) Find an explicit series representation for ∫ f(x) dx from 0 to 1.

(c) Prove that f(x) is differentiable on (0, 1), and …find a series for f '(x) on (0, 1).

Suppose that the power series ∑ (k = 0 to ∞) [(a_k)(x^k)] has a radius of convergence 2. Find the radius of convergence of the series

(a) ∑ (k = 0 to ∞) [(a_k)(x^nk)] where n is a …fixed positive integer.

(b) ∑ (k = 0 to ∞) [(a_k)(x^(k^2)]

Suppose that ∑ (k = 0 to ∞) [(a_k)(x^k)] has radius of convergence R belong to (0, ∞).

(a) Find the radius of convergence of ∑ (k = 0 to ∞) [(a_k)(x^2k)].

(b) Find the radius of convergence of ∑ (k = 0 to ∞) [((a_k)^2)(x^k)].

Find the interval of convergence for the power series ∑ (k = 1 to ∞) [log ((k+1)/k)] * x^k

Show that f(x) = ∑ (k = 1 to ∞) (1/k) sin (x/(k + 1)) converges, pointwise on R and uniformly on each bounded interval in R, to a differentiable function f which satisfies |f(x)| <= |x| and |f ' (x)| <= 1 for all x belong to R.