Theorems on divergent sequences

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

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

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

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

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 +∞

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 +∞

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 +∞

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 -∞

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 -∞

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

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

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