Real Analysis Answers

For all even integral value of n, lim (x+1)^-n

n to ∞

Exist or not

True or false with full explanation

Test whether the series ∞Σn=0 1/(n^5+x^3) converges uniformly or not

A real function f defined on an interval [a,b] with a<c<b where c is a point of the interval, is said to be differentiable at the point x=c if

Applying Cauchy’s mean value theorem to the function f and g defined as f(x)=x2 and g(x)=x for all x∈[a,b], gives

Test whether the series ∞Σn=0 1/(n^5+x^3) converges uniformly or not

Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:

f ={-x², when x ≤ 0.

4-5x , when 0<x≤1

3x-4x², when 1<x≤2

-12x + 2x , when x>2}

Also check whether the function f is derivable at x = 1

Evaluate:

Lim↓ n→∞ [ n/1+n² + n/4+ n² +n/9+ n² +...... +n/2n²]

Show that (1/n²+ n+1)↓n∈N

is a Cauchy sequence.

The product of two divergent sequences is divergent. True or false? Justify.

Give an example of a divergent sequence which has two convergent subsequences.

Justify your claim.