Td2 Suites

S. Jaubert - CFAI-CENTRE

TD2 Suites Numériques 1. Trouver la limite des suites suivantes : a. n

n

u n  ∑ 1  k−1 3

;

vn 

k1

∑ 23  k k3

b. n

k u n  ∑ 5 3k−1 7 k30

n

;

vn 

∑  4  k

k100

c. n

un 

∑ sink k1

n

;

vn 

∑ cosk k1

2. In his book Liber Abaci (book of the Abacus), Leonardo of Pisa, also know as Fibonacci, posed the following question : How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on ? (A History of Mathematics by Carl B. Boyer, Princeton University Press, 1985, page 281). a. Let f n be the number of pairs of rabbits in the nth month. Explain why f 1  1 and f2  1 b. Explain why f n2  f n1  f n for n  1, 2, 3 c. Compute f n for n  3, 4, 5 d. What’s lim n→ f n ? fn e. Compute r n  f n1 for n  1, 2, 3. Do you think lim n→ r n exist ? f. Can you find lim n→ r n exactly ? 3. Etudier les suites suivantes : a. 3u n1 − 1  u n , n ≥ 0 où u 0  1 b. u n2  4u n1 − 4u n , n ≥ 0 où u 0  0 et u 2  5 c. u n1 

u n , n ≥ 0 où u  2 0 3 − 2u n

d. u n1  3u n − 1 , n ≥ 0 où u 0  2 un  1