Real Analysis Answers

((xn,yn) is convergent iff both (xn}) and (yn) are convergent. In fact, for(x0 ,y0) in R2, we have (xn ,yn) converging to (x0,y0) iff xn converges to x0 and yn converges to y0

Given a sequence ((xn,yn)) is R2 .prove that if ((xn,yn)) is bounded ,then (xn) and (yn) are bounded.

find the nth term of the following series 1*2/3^2*4^2 + 3*4/5^2*6^2 + 5*6/7^2*8^2 + ... is convergent.

Show that the sequence (a_n) is bounded iff |a_n| is bounded .

What is the Fourier cosine series for f(x) = 2-π, in the interval 0 ≤ x < 2π?

Show that the sequence En=(1+1/n)^n is bounded and increasing

Let P = (3, 4), ò= (0, 0), P0 = (5,0) be points in R²

equipped with the

Euclidean metric. Let R be the ring given by R ={v∈ R²: 4 < d(ò, v) ≤ 7}.

Which of the following are neighborhoods of P?

R, B1 = B(ò, 4), B2 = B(ò, 6), B3 = R \ B2, B4 = B(ò, 1) ∪ B2, B5 = B(P0,√11).

Let f be a differentiable function on ] [α, β and ]. x ∈[α, β Show that, if 

f ′(x) = 0 and ,0 f ′′(x) > then f must have a local maximum at x.

Show that n=1 to ∞ ∑(-1)^n+1 5/7n+2 is conditionally convergent.

Separate the interval in which the function f defined on R by f(x)=2x3-15x2+36x+5 for all x∈R is increasing