Let converges to ‘l’ and converges to ‘m’. Let converges to s, let k be a non-zero fixed number then converges to ks.Ĥ. if positive terms of convergent series change their sign, then the series will be convergent.ģ. the convergence and divergence of an infinite series is unchanged addition od deletion of a finite number of terms from it.Ģ. Oscillatory series- when Sn does not tends to a unique limit (finite or infinite), then it is called Oscillatory series. The sequence may or may not take the value of the limit. ĭivergent series– when Sn tends to infinity then the series is said to be divergent. A Convergent Sequence is a sequence which becomes arbitrarily close to a specific value, called its 'limit'. Infinite series- If is a sequence, then is called the infinite series.Ĭovergent series – suppose n→∞, Sn→ a finite limit ‘s’, then the series Sn is said to be convergent. The above definition could be made more precise with a more careful definition of a limit, but this would go. Here we can see that, the sequence Sn is divergent as it has infinite limit. A sequence that is not convergent is divergent. As we can see that the sequence Sn is convergent and has limit 1.Įxample-2: consider a sequence Sn= n ² + (-1) ⁿ. Note- a sequence which neither converges nor diverges, is called oscillatory sequence.Ī sequence is null, when it converges to zero.Įxample-1: consider a sequence 2, 3/2, 4/3, 5/4, ……. Oscillatory sequence- when a sequence neither converges nor diverges then it is an oscillatory sequence. That means the limit of a sequence Sn will be always finite in case of convergent sequence.ĭivergent sequence- when a sequence tends to ±∞ then it is divergent sequence. Definition 3.1.2: Convergence A sequence of real (or complex) numbers is said to converge to a real (or complex) number c if for every > 0 there is an integer. Convergent sequence- A sequence Sn is convergent when it tends to a finite limit.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |