Prove that if a series is convergent, then its partial sums, s_n, are bounded.

Thanks in advance. Appreciate it.

Printable View

- Dec 2nd 2008, 04:36 PMthahachainaAnalysis (series)
Prove that if a series is convergent, then its partial sums, s_n, are bounded.

Thanks in advance. Appreciate it. - Dec 2nd 2008, 04:45 PMwatchmath
Well basically you need to show that every convergence sequence is bounded.

If convergence (say to ) then there is N such that for all we have

| .

Hence for . Now for every , so it is bounded. - Dec 2nd 2008, 04:52 PMPlato
- Dec 2nd 2008, 05:51 PMMathstud28
A more direct way would warrant this. Let . So then for to converge to a finite value, let's say , we must have that . But now assuming that we are talking about we have that the defintion of a sequence converging to is for every there exists a such that . Now since are real there exists a number let say such that . So it follows that . Thus it is bounded.