Evaluate
The given limit is: \[ \lim_{n \to \infty} \left[ \frac{n}{1 + n^2} + \frac{n}{4 + n^2} + \frac{n}{9 + n^2} + \dots + \frac{n}{2n^2} \right] \] ________ _______ _____ ______ _________ _________ _________ ______ _________ ____ _________ _____.
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Give an example of a divergent sequence which has two convergent subsequences. Justify your claim.
Check whether the series n is uniformly convergent or not, where . + α ∈
Which of the following statements are true and which are false? Justify your answers with a short proof or a counter-example.
i) If x and y are real numbers such that x < y, then . x2 < y2
ii) For every finite set S, sup S ∈ S
iii) There exists an interval with infimum and supremum equal.
iv) The sequence is unbounded.
v) If a sequence is bounded, then it has at least two convergent subsequences.
Find the following limit.
Test the following series for convergence.
Show that is a Cauchy sequence.
Let f be a differentiable function on [α, β] and x ∈[α, β] Show that, if f ′(x) = 0 and , f ′′(x) > 0then f must have a local maximum at x.