1. Theorems on divergent sequences

2. Theorem 1 If the sequence is increasing and not bounded from above then it diverges to +∞. Illustration =

3. Theorem 2 If the sequence is decreasing and not bounded from below then it diverges to -∞. Illustration =

4. Theorem 3 The sequence diverges to +∞ iff the sequence diverges to -∞. Illustrations = diverges to +∞, while = diverges to -∞

5. Theorem 4 If the sequences and diverge to +∞, then the sequences and diverge to +∞. Illustrations and diverge to +∞ → and diverge to +∞

6. Theorem 5 If the sequences and diverge to -∞, then the sequences diverges to -∞, while the sequence diverges to +∞. Illustrations and diverge to -∞ → diverges to -∞, While diverge to +∞

7. Theorem 6 If the sequence diverges to +∞ and the sequence is bounded then the sequence diverges to +∞ Illustration diverges to +∞ and is bounded → diverges to +∞

8. Consequence of Theorem 6 If the sequence diverges to +∞ and the sequence is convergent then the sequence diverges to +∞ Illustrations diverges to +∞ and is convergent → diverges to +∞

9. Theorem 7 If the sequence diverges to -∞ and the sequence is bounded then the sequence diverges to -∞ Illustration diverges to -∞ and is bounded → diverges to -∞

10. Consequence of Theorem 7 If the sequence diverges to -∞ and the sequence is convergent then the sequence diverges to -∞ Illustration diverges to -∞ and is convergent → diverges to -∞

11. Example (1) Find the limit of the sequence, if converges

12. Example (2) Find the limit of the sequence, if converges

13. Example (3) Find the limit of the sequence, if converges