Check whether or not the sequence , defined by:
and is convergent.
The sequence is defined by (a_1=2) and (a_{n+1}=2-\dfrac{1}{a_n+2}) for all (n\ge1). First compute a few terms to observe its behavior. Since (a_1=2), we have [ a_2=2-\frac{1}{2+2}=2-\frac14=\frac74, ] [ a_3=2-\frac{1}{\frac74+2}=2-\frac{1}{\frac{15}{4}}=2-\frac{4}{15}=\frac{26}{15}. ] These __________ _______ _____ _____ _______ _____.
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फलन f की पर असांतत्य के प्रकार की जाँच कीजिए, जहाँ
है।
Check whether or not the sequence , defined by:
and is convergent.
जाँच कीजिए कि पूर्णांकों का समुच्चय गणनीय है या नहीं।
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
.
यदि अनंत श्रेणी अभिसारी है, तो दिखाइए कि
होती है।
Let f be a differentiable function whose derivative never vanishes on [a, b]. Show that f is either strictly increasing or strictly decreasing.
उपयुक्त प्रति उदाहरण देकर इस कथन को असिद्ध कीजिए:
प्रत्येक तथा
के लिए
है।