Exercice 3: Pour chacune des suites suivantes, étudier sa nature, son sens de variation et conjecturer sa convergence ; préciser de plus si elle est, ou non, majorée ou minorée:
a) $\quad u_n=n^2-n$
b) $\quad u_n=\frac{n+2}{n-3}$
c) $\quad u_n=-4 \times\left(\frac{2}{3}\right)^n$
d) $\quad u_n=1-\frac{3}{2} n$
e) $\quad u_n=\sin \left(n \frac{\pi}{2}\right)-3$
f) $\quad u_n=\frac{2^n-3}{2^n+1}$
Exercice 4:
Answer (1)
Sequence a) u n = n 2 − n increases, diverges to ∞ , and is minorized by 0.
Sequence b) u n = n − 3 n + 2 decreases for 3"> n > 3 , converges to 1, and is bounded below.
Sequence c) u n = − 4 × ( 3 2 ) n increases, converges to 0, is majorized by 0, and minorized by − 3 8 .
Sequence d) u n = 1 − 2 3 n decreases, diverges to − ∞ , and is majorized by − 2 1 .
Sequence e) u n = sin ( n 2 π ) − 3 oscillates, does not converge, is majorized by -2, and minorized by -4.
Sequence f) u n = 2 n + 1 2 n − 3 increases, converges to 1, is minorized by − 3 1 , and majorized by 1.
Explanation
Problem Analysis We are given six sequences and need to analyze their nature, sense of variation, convergence, and whether they are majorized or minorized.
Sequence a a) u n = n 2 − n This is a polynomial sequence. As n increases, u n also increases. Thus, it is increasing. To check convergence, we find the limit as n approaches infinity: lim n → ∞ ( n 2 − n ) = ∞ . Therefore, the sequence diverges to infinity. Since the sequence is increasing and starts at u 1 = 0 , it is minorized by 0 but not majorized.
Sequence b b) u n = n − 3 n + 2 To determine the sense of variation, we can analyze the difference between consecutive terms or consider the derivative of the corresponding continuous function. Let's analyze the limit first. lim n → ∞ n − 3 n + 2 = 1 . Thus, the sequence converges to 1. To find the variation, let's look at the first few terms: u 4 = 6 , u 5 = 2 7 = 3.5 , u 6 = 3 8 ≈ 2.67 . It appears to be decreasing for 3"> n > 3 . It is neither majorized nor minorized as it approaches 1. For n < 3 the terms are negative, so it is bounded below.
Sequence c c) u n = − 4 × ( 3 2 ) n This is a geometric sequence with a common ratio of 3 2 . Since ∣ 3 2 ∣ < 1 , the sequence converges to 0. Since the common ratio is positive, and the initial term is negative, the sequence is always negative and increasing towards 0. It is majorized by 0 and minorized by the first term u 1 = − 4 × 3 2 = − 3 8 .
Sequence d d) u n = 1 − 2 3 n This is an arithmetic sequence with a common difference of − 2 3 . Therefore, the sequence is decreasing. The limit as n approaches infinity is lim n → ∞ ( 1 − 2 3 n ) = − ∞ . Thus, the sequence diverges to negative infinity. The sequence is majorized by the first term u 1 = 1 − 2 3 = − 2 1 , but not minorized.
Sequence e e) u n = sin ( n 2 π ) − 3 The sine function oscillates between -1 and 1. Therefore, the sequence oscillates between -4 and -2. Since it oscillates, it does not converge. It is majorized by -2 and minorized by -4.
Sequence f f) u n = 2 n + 1 2 n − 3 To determine the sense of variation, we can analyze the difference between consecutive terms or consider the derivative of the corresponding continuous function. Let's analyze the limit first. lim n → ∞ 2 n + 1 2 n − 3 = 1 . Thus, the sequence converges to 1. To find the variation, let's look at the first few terms: u 1 = 2 + 1 2 − 3 = − 3 1 , u 2 = 4 + 1 4 − 3 = 5 1 , u 3 = 8 + 1 8 − 3 = 9 5 . It appears to be increasing. The sequence is minorized by − 3 1 and majorized by 1.
Final Answer In summary: a) u n = n 2 − n : Increasing, diverges to infinity, minorized by 0. b) u n = n − 3 n + 2 : Decreasing for 3"> n > 3 , converges to 1, bounded below. c) u n = − 4 × ( 3 2 ) n : Increasing, converges to 0, majorized by 0, minorized by − 3 8 .
d) u n = 1 − 2 3 n : Decreasing, diverges to negative infinity, majorized by − 2 1 .
e) u n = sin ( n 2 π ) − 3 : Oscillating, does not converge, majorized by -2, minorized by -4. f) u n = 2 n + 1 2 n − 3 : Increasing, converges to 1, minorized by − 3 1 , majorized by 1.
Examples
Understanding the behavior of sequences is crucial in many areas of mathematics and its applications. For example, in finance, the sequence of returns on an investment can be analyzed to determine its long-term performance and risk. Similarly, in physics, the motion of an object can be described by a sequence of positions at different times. Analyzing the convergence, variation, and boundedness of these sequences helps in making predictions and understanding the underlying phenomena. For instance, determining if a sequence of approximations in a numerical method converges is essential to ensure the accuracy of the solution.
