Real Analysis Answers

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

Give an example of a divergent sequence which has two convergent sequences. Justify  your claim

lim n tend to infinity [1/√2n-1 + 1/√4n-2² +1/√6n-3²+ .......+ 1/n] = π/2

Check whether the intervals [2,5] and [7,10] are equivalent or not

Show that the sequence (an), where an= n/n^2+4 is monotonic. Is (an) a Cauchy sequence? Justify your answer

The function: f:[-1,3] →R defined by:

f(x)= 3x+1/x^2+4 is uniformly continuous on [-1,3].

True or false with full explanation

The series

1/3+ 1/7+ 1/11 + 1/15+.....

is a convergent series.

True or false with full explanation

An integrable function can have finitely many points of discontinuous.

True or false with full explanation

The set Q of a rational numbers is a closed subset of R.

True or false

Test the following series for convergence,

Σ n x^n-1 , x > 0