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# Functions : Convergence and Limits

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Let f be a real function defined by .

1) Evaluate f'(x), f''(x), f(0). Show that f has exactly two roots and , with . Find an interval of two consecutive real numbers within which the roots must lie.
From now on, let us denote and these two (closed) intervals.
2) Let be the sequence defined by and

a) Show that for all whole natural numbers .
b) Show that if the sequence is convergent, its limit is .
c) Evaluate g'(x). Show that the sequence is convergent and give a sufficient number of iterations N such that the approximation error is no more than .
3) Let and let be the sequence defined by Newton's method starting with . Show that this sequence is convergent. Give its limit. Evaluate .

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