Check whether the set of integers is countable or not.
See Answer →Give one example for each of the following. Justify.
(i) A set which is neither open or closed.
(ii) A set which has no limit point.
State Bolzano-Weierstrass theorem. Use it to check whether the set:
has a limit point.
Disprove the statement:
by providing a suitable counter-example.
Prove that every convergent sequence is Cauchy.
See Answer → Prove that .
Discuss the convergence of the p-series by using the integral test.
Suppose that f and g are in R[a, b]. Then show that the function f + g is in R[a, b], and
Examine the type of discontinuity of the function at where
Show that the function defined as
is continuous on its domain, but not uniformly continuous there. Will the function be uniformly continuous on
for any fixed c > 0?
Show that the set:
is not closed.
Find the radius of convergence of the series , where
.
Test the convergence of the series: .
Check whether or not the sequence , defined by:
and is convergent.
State Cauchy’s second theorem on limits. Use it to show that:
Using mathematical induction, prove that the sum of the squares of first 'n' natural numbers is .
State whether the following statements are true or false. Justify your answers with a short proof or a counterexample.
i) The negation of is
.
ii) is countable.
iii) The equation has a real root between -2 and 1.
iv) Every increasing sequence has a convergent subsequence.
v) The function defined as
has a removable discontinuity.
vi) An integrable function can have finitely many points of discontinuity.
vii) The series is divergent.
viii) The function f defined by , has local maximum at
.
ix) Every constant function on [a, b] is Riemann Integrable.
x) The function defined by
is differentiable at
.
यदि अनंत श्रेणी अभिसारी है, तो दिखाइए कि
होती है।
मान लीजिए कि f एक अवकलनीय फलन है, जिसका अवकलज अंतराल [a, b] पर शून्य नहीं होता। तब दिखाइए कि f या तो सर्वथा वर्धमान है अथवा सर्वथा ह्रासमान है।
See Answer →