Real Analysis Answers

Q. If f, g is Reimann integrable on [a, b] then fg is reimann integrable on [a, b].

Q. Define the following terms
Series
Conditionally convergent series
Completeness property

Compute the Riemann integral of the function f (x) = x on the interval [−1, 1].

Verify Inverse function theorem for finding the derivative at a point y of the 0
domain of the inverse function of the function f (x) = cosx, x ∈[0,π].Hence, find
the derivative of the inverse function aty

Verify the second mean value theorem of integrability for the functions f and g
defined on [1,2] by f (x) = 3x and g(x) = 5x.

Q: show that if zn=(an+bn)1/n where 0<a<b, then lim(zn)=b.

Q: If a>0, b>0, show that lim(√((n+a)(n+b))-n)=(a+b)/2

Q: If 0<a<b, determine lim((a^(n+1)+b^(n+1))/(a^n+b^n ))

Q: Determine the following limits:
(a) lim((3√n)1/2n), (b) lim((n+1)1/ln(n+1)).

Q. Let yn=√(n+1)-√n for nϵN. Show that (yn) and (√nyn) converge. Find their limits.