Show that the set:
is not closed.
Consider the set [ S = \left{ \frac{1 + (-1)^n}{2} + \frac{1}{n} \mid n \in \mathbb{N} \right}. ] For even (n = 2k), (\frac{1 + (-1)^n}{2} = ______ ___ ______ ______ __________ _________ _______.
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श्रेणी , की अभिसरण त्रिज्या ज्ञात कीजिए। जहाँ
है।
Disprove the statement:
by providing a suitable counter-example.
Examine the type of discontinuity of the function at where
सिद्ध कीजिए कि प्रत्येक अभिसारी अनुक्रम कौशी होता है।
Find the radius of convergence of the series , where
.
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
.
सिद्ध कीजिए कि यह समुच्चय संवृत नहीं है।