The document provides a comprehensive overview of concepts related to infinite sequences and series in calculus, including definitions, properties, and various tests for convergence or divergence. Key topics covered include the limit of a sequence, monotonic and bounded sequences, absolute and conditional convergence, and Taylor and Maclaurin series. Additionally, it outlines guidelines for testing series and describes the convergence of power series and their properties.

1. PRE-MIDTERM
INFINITE SEQUENCES & SERIES
CALCULUS III
PREPARED BY: T. CARL

2. DEFINITION OFTHE LIMIT OF A SEQUENCE

3. LIMIT OF A SEQUENCE

4. PROPERTIES OF LIMITS OF SEQUENCES

5. SQUEEZE THEOREM FOR SEQUENCES

6. ABSOLUTE VALUE THEOREM

7. DEFINITION OF A MONOTONIC SEQUENCE

8. DEFINITION OF A BOUNDED SEQUENCE

9. BOUNDED MONOTONIC SEQUENCES

10. DEFINITIONS OF CONVERGENT AND
DIVERGENT SERIES

11. CONVERGENCE OF A GEOMETRIC SERIES

12. PROPERTIES OF INFINITE SERIES

13. LIMIT OF NTH TERM OF A CONVERGENT SERIES

14. NTH-TERMTEST FOR DIVERGENCE

15. THE INTEGRAL TEST

16. CONVERGENCE OF P-SERIES

17. DIRECT COMPARISON TEST

18. LIMIT COMPARISONTEST

19. ALTERNATING SERIESTEST

20. ALTERNATING SERIES REMAINDER

21. ABSOLUTE CONVERGENCE

22. DEFINITIONS OF ABSOLUTE AND
CONDITIONAL CONVERGENCE

23. RATIO TEST

24. ROOT TEST

25. GUIDELINES FOR TESTING A SERIES FOR
CONVERGENCE OR DIVERGENCE

26. SUMMARY OFTESTS FOR SERIES

27. SUMMARY OFTESTS FOR SERIES (CONT’D)

28. DEFINITIONS OF NTH TAYLOR POLYNOMIAL
AND NTH MACLAURIN POLYNOMIAL

29. TAYLOR'STHEOREM

30. DEFINITION OF POWER SERIES

31. CONVERGENCE OF A POWER SERIES

32. PROPERTIES OF FUNCTIONS DEFINED BY
POWER SERIES

33. OPERATIONSWITH POWER SERIES

34. THE FORM OF A CONVERGENT POWER SERIES

35. DEFINITIONS OF TAYLOR AND MACLAURIN SERIES

36. CONVERGENCE OFTAYLOR SERIES

37. GUIDELINES FOR FINDING A TAYLOR SERIES

38. POWER SERIES FOR ELEMENTARY FUNCTIONS

39. REFERENCE: