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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}

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
The function f (x) = x2 + x , is differentiable at x = −1
Every subsequence of the sequence

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 ?
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
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.
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