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1) EVALUATE

Limit {(2+x)^5/4-(2)^5/4}/{(2+x)^2/3-(2)^2/3}

x tends to zero

Limit {(2+x)^5/4-(2)^5/4}/{(2+x)^2/3-(2)^2/3}

x tends to zero

Find whether the following sequences converge or not

A) {2+(-1)^n}

B)(4n^3+n)/(2n^3+7n)

A) {2+(-1)^n}

B)(4n^3+n)/(2n^3+7n)

a) Every infinite set is an open set.

b)A necessary condition for a function f to be integrable is that it is continuous.

true or false

b)A necessary condition for a function f to be integrable is that it is continuous.

true or false

The function f (x) = x2 + x , is differentiable at x = −1

Every subsequence of the sequence

(1/n^2)

is convergent

(1/n^2)

is convergent

Show that n!<n^n, n>1, by induction.

d^2u/dx^2 =6x+12y^2 subject to u(1,y) = y^2-2y ,u(x,t)=5x-5

Qno.1) a) check whether the following sequences are cauchy sequence or not ?

I) (n+1/n) ii) (1+1/2!+.....+1/n!)

iii) (-1)^n iv) (n+ (-1)^n/n)

b) let x1<x2 be arbitrary real number and Xn= 1/2(Xn-2+Xn-1) for n>2. Show that (Xn) is convergent ? What is it limit ?

I) (n+1/n) ii) (1+1/2!+.....+1/n!)

iii) (-1)^n iv) (n+ (-1)^n/n)

b) let x1<x2 be arbitrary real number and Xn= 1/2(Xn-2+Xn-1) for n>2. Show that (Xn) is convergent ? What is it limit ?

Qno.1) Establish the convergence or divergence of the sequence (Xn) such that

I) Xn= 1/1^2+1/2^2+1/3^2+.......1/n^2 for all n belongs to N.

ii) Xn= (1+1/n)^n+1 iii) Xn= (1+1/n+1)^n

Iv) Xn= (1-1/n)^n

I) Xn= 1/1^2+1/2^2+1/3^2+.......1/n^2 for all n belongs to N.

ii) Xn= (1+1/n)^n+1 iii) Xn= (1+1/n+1)^n

Iv) Xn= (1-1/n)^n

Qno. 3) I) let X1=8 and Xn+1=1/2Xn+2 for all n belongs to N .show that X is bounded and monotone.also find the limit.

ii) let X1>1 and Xn+1=2-1/Xn for all n belongs to N. Show that (Xn) is bounded and monotone . Find the limits.

ii) let X1>1 and Xn+1=2-1/Xn for all n belongs to N. Show that (Xn) is bounded and monotone . Find the limits.