Show that
We are asked to show that: $$ e^{-x} < 1 - x + \frac{x^2}{2}, \quad x > 0 $$ Consider the Taylor series expansion of $e^{-x}$ about ______ _________ _______ _________ _____.
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Show that [a,∞)is closed set.
Prove that the sequence (an)n∈N, where an = 32 / x2 + 22 , is Cauchy.
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^~qis p→q.
iii) -1 is a limit point of the interval ]-2,1].
iv) The necessary condition for a function f to be integrable is that it is continuous.
The function f: R→R defined by f(x) = |x-2|+|3-x is differentiable at x = 5.
Let x and y be two real numbers such that x < y.Show that there exists an irrational number λ such that
x < λ < y.
निम्नलिखित कथन तथा उनके निषेध को तर्कसंगत प्रमात्रकों का प्रयोग करते हुए आप किस प्रकार प्रस्तुत करेंगे? निषेध को शब्दों में भी दीजिए।
निम्नलिखित श्रेणियों के अभिसरण की जाँच कीजिए :
(i)
(ii)
दिखाइए कि प्रत्येक [a,∞) एक विवृत समुच्चय है।