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Suppose {p} is a cauchy sequence in a metric space X and some subsequence {p} converges to a point a belongs to X . Prove that the full sequence {p} converges to a .

Show that limit n approaches to infinity (xe^-nx)=0 for x € real number , x>0

Q. ∫_(-∞)^∞▒e^(〖-x〗^2 ) dx=?

a. π/2

b. π

c. √π

d. 2√π

a. π/2

b. π

c. √π

d. 2√π

Q. Choose the correct answer.

Q. The series ∑_(n=1)^∞▒〖(-1)〗^(n+1) n/(n^2+π) is

a. conditionally convergent for n>√π

b. absolutely convergent for n>π^2

c. divergent for n>0

d. none of the above

Q. The series ∑_(n=1)^∞▒〖(-1)〗^(n+1) n/(n^2+π) is

a. conditionally convergent for n>√π

b. absolutely convergent for n>π^2

c. divergent for n>0

d. none of the above

Q. Choose the correct answer.

Q. Which of the following is the sum of infinite series ∑_(n=0)^∞▒〖(-1)〗^n 3^n r^n?

a. 1/(1+3r) for -1/3<r<1/3

b. . 1/(1-3r) for -1/3<r<1/3

. 3/(1-3r) for -1/3<r<1/3

d. ∞

Q. Which of the following is the sum of infinite series ∑_(n=0)^∞▒〖(-1)〗^n 3^n r^n?

a. 1/(1+3r) for -1/3<r<1/3

b. . 1/(1-3r) for -1/3<r<1/3

. 3/(1-3r) for -1/3<r<1/3

d. ∞

Q. Choose the correct answer.

Q. If a_n=∑_(l=1)^n▒〖(□(1/l))〗, then which of the following statement si true about the sequence {a_n} ?

a. Chauchy sequence.

b. Convergent sequence

c. Not a Cauchy sequence

d. Every subsequence of {a_n} is convergent.

Q. If a_n=∑_(l=1)^n▒〖(□(1/l))〗, then which of the following statement si true about the sequence {a_n} ?

a. Chauchy sequence.

b. Convergent sequence

c. Not a Cauchy sequence

d. Every subsequence of {a_n} is convergent.

Q. Choose the correct answer.

Q. Which of the following statement is true for sequence {(〖-1)〗^(n-1)}?

a. The sequence is bounded

b. The sequence is increasing

c. The sequence is decreasing

d. The sequence is neither increasing nor decreasing

Q. Which of the following statement is true for sequence {(〖-1)〗^(n-1)}?

a. The sequence is bounded

b. The sequence is increasing

c. The sequence is decreasing

d. The sequence is neither increasing nor decreasing

For each x ∈ R, let us denote by C(x) the least integer greater than or equal to x.

For example, C(1) = 1, C(−√2) = −1. In other words, C(x) is the unique integer

satisfying C(x) − 1 < x ≤ C(x).

(1) Draw the graph of the function C(x) for x ∈ [−2, 2].

(2) Prove that C(x) is continuous at all non-integer points of R.

(3) Prove that C(x) is discontinuous at all integer points of R.

For example, C(1) = 1, C(−√2) = −1. In other words, C(x) is the unique integer

satisfying C(x) − 1 < x ≤ C(x).

(1) Draw the graph of the function C(x) for x ∈ [−2, 2].

(2) Prove that C(x) is continuous at all non-integer points of R.

(3) Prove that C(x) is discontinuous at all integer points of R.

Prove that the sequence {f_n(x)} where f_n(x)=nx/1+n^2x^2

is not uniformly convergent in [-1,1].

is not uniformly convergent in [-1,1].

Show that the function f:[2,3]→R defined by :

f(x)={0 if x is rational {1 if x is irrational

is discontinues and not integrable over [2,3].Does it imply that every discontinues function is non-integrable?Justify your answer.

f(x)={0 if x is rational {1 if x is irrational

is discontinues and not integrable over [2,3].Does it imply that every discontinues function is non-integrable?Justify your answer.