Question

In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if it is bounded. The readers who are interested in the proof of the above theorem can refer in the bibliography. The theorem is usually used to prove other tmy s on conver of a sequence or series. Example 6.10 Use Theorem 6.9 to show that (n/(n + 1) converges. Solution We have shown in Example 6.8 that the sequence is monotonic incresin Example 6.9 it is bounded. By Theorem 6.9, the sequence in/(n+1) converge d n g, andin Exercise 6.5 In questions 1-8, use Theorem 6.9 to determine whether the sequence converges or diverges. 2. 2n +1 1. 4. en 5. (n2) 7. n + In n l + en 9. Given that -1)"/n3 is a sequence. (G) Show that the sequence is not monotonic. (i) Show that the sequence converges. (iii) Is this result contradicts Theorem 6.9?

Answer 1

let in Codan 3 3 h how. lel é z be any. Then mzmo, 1an-ole. Hence li'm au n! Eza , then, tet di 느그 Hence lim 0. NE