श्रेणी के अभिसरण की जाँच कीजिए।
दी गई श्रेणी [ \sum_{n=1}^{\infty}\frac{(-1)^{,n+1}n}{n^3+1} ] है। इसके अभिसरण की जाँच के लिए पहले इसके परिमाण की _______ ____ _________ ____ ___ _______ _______ _______ __________ ___.
__________ _________ _____ ______ __________ ____ _____ ____ ___.
______ _______ ______ ________ _____ ______ ____ _____ _____ ____ ________.
___ ____ _____ ____ __________ _______ __________ ____ ________ _____ _______.
___ ________ ____ ________ _____ __________ ______ ____.
____ ________ ______ ______ _________ ___ __________ _______ ______ ________ _____.
_________ ___ _______ _____ _____ ____ ____ __________.
______.
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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?
Check whether the set of integers is countable or not.
State Cauchy’s second theorem on limits. Use it to show that:
Examine the type of discontinuity of the function at where
If an infinite series converges, then show that
.
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
.
जाँच कीजिए कि अनुक्रम , जिसे इस प्रकार से परिभाषित किया जाता है:
, और
.
अभिसारी है या नहीं।