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