Real Analysis Answers

Define uniform convergence of a sequence of functions. Discuss the uniform convergence of the sequence nx^2+1/nx+1 the interval  [1. 2]

a differentiable real valued function f has at the point (1, 2), directional derivatives +2 in the direction toward (2, 2) and -2 in the direction toward (1, 1). determine the gradient vector at (1, 2) and compute the directional derivative in the direction toward (4, 6).

Show that sinx and cosx are a bound variation on a finite interval

Is f(x)=1/x^2/3 in L₃[1 2] ?

Is 1/x^2/3 continuous on [1 2] ?

a differentiable real valued function f has at the point (1, 2), directional derivatives +2 in the direction toward (2, 2) and -2 in the direction toward (1, 1). determine the gradient vector at (1, 2) and compute the directional derivative in the direction toward (4, 6).

Consider the sequence {xn}, which is defined by

x1 = 1, xn+1 = xn +

1/

x1 + x2 + · · · + xn

, n ∈ N.

Does {xn} converge? Justify your answer.

Let Pun be a series of arbitrary terms. For all n ∈ N, let pn =

1

2

(un + |un|) and qn =

1

2

(un − |un|). Show

that

(a) If Pun is absolutely convergent, then both Ppn and Pqn are convergent.

(b) If Pun is conditionally convergent, then both Ppn and Pqn are divergent.

Give an example of a function in L₃[1 2] that is not in C [1 2]

Consider the sequence {xn}, which is defined by

x1 = 1, xn+1 = xn +

1

x1 + x2 + · · · + xn

, n ∈ N.

Does {xn} converge? Justify your answer.

(2)

? ? ? ? ?