Limit of a Sequence

1. Convergence of a Sequence
   A sequence is said to converge to a value if, as becomes infinitely large, the value of gets arbitrarily close to . In this case, we say that the sequence converges to , and is called the limit of the sequence. This is expressed as:
   For example, in the graph below, the sequence converges to , so:
   
   
   In another example, the sequence converges to , so:
   
   
   For a constant sequence where for all , it converges to , so:
   
   
   
   
2. Divergence of a Sequence
   A sequence is said to diverge if it does not converge to any specific value. There are several cases of divergence:
   
   
1. Divergence to Positive Infinity
   If the terms of a sequence grow without bound as increases, we say that the sequence diverges to positive infinity, expressed as:
   or as
   
   For example, .
   
   
   
   
   
   
2. Divergence to Negative Infinity
   If the terms of a sequence become increasingly negative as increases, we say that the sequence diverges to negative infinity, expressed as:
   or as
   
   For example, .
   
   
   
   
   
   
3. Non-convergence without Divergence to Infinity
   Some sequences neither converge to a finite value nor diverge to infinity. In such cases, the terms may oscillate between values. For example:
   The sequence with general term alternates between and .
   
   
   
   The sequence with general term increases in magnitude while alternating between positive and negative values.
   
   
   
   In both cases, the sequences do not converge to a finite value, nor do they diverge to positive or negative infinity.