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
.
i) False. The negation of an equation (f(x) = g(x)) is (f(x) \neq g(x)), not the same equation. ii) True. The set of rational numbers, often represented by "equation" in countability ________ ______ __________ _____ _______ ____ ______ ________ ________ _______.
______ ________ _______ ___ ______ ____ _____ _________ _______ _________.
__________ ____ _______ ___ ___ _____ _____ ____ ______ _______ ____ __________.
______ ___ __________ _________ ______ _________ __________ _____ ___.
_____ __________ ________ ________ ______ _________ ____ _______ __________ ______ _______.
______ _____ __________ _______ _______ ____ _______.
____ _______ _______ ______ _______.
_____ ____ __________ _______ ______ ________ __________ ______ ________ ______ ______ ______.
_______ ______ _________ __________ ________ ____ ___.
_______ _______ ________ ____ ___ ______ _______ ___.
_____ ______ ___ _____ _______ ______ _________ _______ __________ _______.
____ ___ ______ ______ __________.
______ ___ ____ _______ _____ __________ _________ __________ __________ ______ _____.
_____ ____ _______ ____ _______ ___ __________ ______ _______ _______ _______ ___.
___ _________ _________ ______ ____ _______ ____.
_____ _____ ________ ________ _____ _____ ______ _________ _________ __________ ______ ________.
____ _______ __________ ______ _______ ________ _____.
______ _______.
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निम्नलिखित में से प्रत्येक के लिए एक - एक उदाहरण दीजिए। अपने उत्तर का कारण बताइए।
i. एक ऐसा समुच्चय जो न तो विवृत है और न संवृत है।
ii. एक ऐसा समुच्चय जिसका कोई सीमा बिंदु नहीं है।
श्रेणी , की अभिसरण त्रिज्या ज्ञात कीजिए। जहाँ
है।
Check whether the following sets are open or closed or neither:
(i)
(ii)
Discuss the convergence of the p-series by using the integral test.
जाँच कीजिए कि निम्नलिखित समुच्चय विवृत हैं या संवृत हैं अथवा न तो विवृत हैं और न ही संवृत।
i.
ii.
Let f be a differentiable function whose derivative never vanishes on [a, b]. Show that f is either strictly increasing or strictly decreasing.
समाकल परीक्षण का प्रयोग करके p − श्रेणी , जहाँ p > 0, के अभिसरण पर चर्चा कीजिए।