BSC 3rd SEM MATHS SERIES NOTES
 |
| WAIT FOR15 Sec & Get DOWNLOAD Link |
Introduction
A Series the sum of the terms Informally speaking the sum of the terms of a sequence.
Finite sequence and series have defined first and last terms, where as Infinite series and sequence continue indefinitely.
• Sequence of partial sums
Let Ean = Art A2+ A3 +----+ant be a given infinite series. And S1, S2, S3, Sn, are the different partial sums called sequence of Ean then the sequence {Snt is of partial sums of Σαπ Thus, every series Ian, these corresponding a unique sequence Sn of it's partial sum
- Nature and Behaviours of Infinite Series
The series Zan is said to converge, diverée or oscillate according as the sequenice (Sn) of. Partial sums of the series Zan converges. diverges or oscillates.
1) Convergent Series -
The seties, to converge, if the sequence Man is said (Sn) of its partial sums converges to a real number I i.e. Lim SmI (finite & unique).
2) Divergent Series -
The series Ean is said to diverge to infinity And if the sequence {Sn} diverses to infinity .And the Ean is said to diverge to negative infinity if the sequence {Sn} deverges to negative infinity.
3) Oscillatory Series -
If the series Ean neither converges nor diverges to infinity or negative infinity,it is said to oscillate finitely,
★ If the sequence {Sn} is bounded then it oscillates infinitely.
★ If the sequence {Sn} is unbounded then it oscillates infinitely.
• CAUCHY'S ROOT TEST:
Statement: If Σαn is a positive term series and lim (an)^1/n =l then Σαn is.
(i) Convergent if l < 1
(ii) Divergent if l > 1
(iii) Root test fails for l =1
Proof: Given that Σαn is a positive thought series and lim (an)^1/n = l
- Conditionally convergent series:
The series san is Σαn is said to be conditional
Convergent or semiconvergent if Σαn is convergent but Σαn is not no convergent.
• Alternating series:
The Series with alternatively positive & negative terms is called alternating series.
------
ETC.....
