Give an example of a divergent sequence which has two convergent subsequences. Justify your claim.
Consider the sequence \( a_n = (-1)^n \).
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Give an example of a divergent sequence which has two convergent subsequences. Justify your claim.
Check whether the series is uniformly convergent or not, wher
Using Weiestrass M-test, show that the following series converges uniformly.
Which of the following statements are true or false? Give reasons for your answers in the form of a short proof or counter-example, whichever is appropriate:
i) Every infinite set is an open set.
ii) The negation of p∧ ~ q is p → q.
iii) −1is a limit point of the interval ]−2 ,1],
iv) The necessary condition for a function to be integrable is that it is continuous.
v) The function defined by
is differentiable at
Test the following series for convergence.
(i)
(ii)
d) Show that is a Cauchy sequence.
Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:
Also check whether the function f is derivable at x = .1