Real Analysis Answers

suppose that $f:[0,1]\rightarrow r$ is continous on [0,1] and differentiable on (0,1) with f(0)=0 anf f(1)=10.prove that there exist 100 distinct points x_k belonging to (0,1) such that summation from k=1 to 100 of 1/f'(x k)=1000

Evaluate

lim 3nΣr=1 n^2/(4n+r)^3

n→∞

Show that the series 8/(n⁴+x⁴) is uniformly convergent for all real values of x. n=1 to infinity .

If lim f(x) and lim g(x) exist as x approaches a then lim [ f(x) / g(x) ] = lim f(x) / lim g(x) as x approaches a. true or false

Let (a ) be any sequence.Show that

lim[ n/1+n2+ n/4+n2+n/9+n2+.....+n/2n2

2000for 5%years with interest rate 12% compounded montly

Is (3x4)/5²+(5x6)/7²+(7x8)/9² +..... series convergence or not

Evaluate:

Limit x tends to infinity( n/1+n^2 + n/4+n^2 + n/9+n^2 +.......n/2n^2 )

Find the value of m for which lim x->infinity

(x+1)(2x-3)(2-3x)/(4-x+mx³) exists.