Real Analysis Answers

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.

Qno.2) Establish either the sequence (X=Xn ) converges or diverges where
I) Xn=n/n+1 ii) Xn= (_1^n)n/n+1 iii)Xn= n^2/n+1
B) find the limits of the sequences
I) (2+1/n)^2 ii) (-1)^n/ n+2. iii) (n)^1/2-1/(n)^1/2+1 iv)n+1/n(n)^1/2
V)a^n+1+b^n+1/a^n+b^n for 0<a<b .

Q no. 1)Use definition of the limit of a sequence to establish the following limits
A) lim (n/n^2+1)=0
B) lim(2n/n+1)=2
C) lim(n^2-1/2n^2+3)=1/2
D) show that
I) lim (1/3^n)=0 ii) lim( 2^n/n!)=0 iii) lim((2n)^1/n)=1

Show that 8^1/2 is not an integer

If ab belongs to R if -(-a)=a prove ??
ii) -1(a)=-a
iii) -(a+b)=(-a)+(-b)
iv) -(a)(-b)=ab
Question: Find real number x such that |x+1|+|x-2|<7??

If ab belongs to R if a+b=0 then a=-b???