Numerical series

Learning tasks

A) Convergence and divergence of a series
I) Series\(\virgule\) partial sums\(\virgule\) sum\(\virgule\) remainder
1) Computing partial sums and remainders	Ex 1 Ex 2
II) Linearity of the sum of convergent series
2) Applying linearity	Ex 3
III) Necessary condition\(\virgule\) gross divergence
3) Recognising gross divergence	Ex 4 Ex 5 Ex 6 Ex 7
IV) Sequence-series link
4) Studying a sequence via its telescoping series	Ex 8 Ex 9 Ex 10 Ex 11
V) Geometric series and the complex exponential
5) Recognising geometric series	Ex 12 Ex 13 Ex 14 Ex 15
B) Series of positive terms
I) Convergence criterion for positive-term series
6) Showing convergence by direct majoration of partial sums	Ex 16
II) Term-by-term comparison
7) Determining the nature of a positive-term series by direct majoration or comparison	Ex 17 Ex 18 Ex 19 Ex 20
III) Comparison by equivalent and by \(O\)
8) Determining the nature via equivalent or \(O\)	Ex 21 Ex 22 Ex 23 Ex 24
IV) Series-integral comparison and Riemann series
9) Determining the nature by comparison to an integral --- Riemann series	Ex 25 Ex 26 Ex 27 Ex 28
C) Absolute convergence
I) Definition and main theorem
10) Showing absolute convergence	Ex 29 Ex 30
II) Comparison by \(O\) for sign-variable series
11) Reducing a sign-variable series to a positive reference	Ex 31 Ex 32 Ex 33
D) Alternating-series criterion
I) Alternating-series criterion and applications
12) Applying the alternating-series criterion	Ex 34 Ex 35 Ex 36 Ex 37
E) Application\(\virgule\) Stirling's formula
I) Stirling's formula via the sequence-series link and Wallis
13) Applying Stirling	Ex 38 Ex 39 Ex 40 Ex 41

Lesson
Text book

Exercises Correction
A) Convergence and divergence of a series
    I) Series\(\virgule\) partial sums\(\virgule\) sum\(\virgule\) remainder
      1) Computing partial sums and remainders Ex 1 Ex 2
    II) Linearity of the sum of convergent series
      2) Applying linearity Ex 3
    III) Necessary condition\(\virgule\) gross divergence
      3) Recognising gross divergence Ex 4 Ex 5 Ex 6 Ex 7
    IV) Sequence-series link
      4) Studying a sequence via its telescoping series Ex 8 Ex 9 Ex 10 Ex 11
    V) Geometric series and the complex exponential
      5) Recognising geometric series Ex 12 Ex 13 Ex 14 Ex 15
B) Series of positive terms
    I) Convergence criterion for positive-term series
      6) Showing convergence by direct majoration of partial sums Ex 16
    II) Term-by-term comparison
      7) Determining the nature of a positive-term series by direct majoration or comparison Ex 17 Ex 18 Ex 19 Ex 20
    III) Comparison by equivalent and by \(O\)
      8) Determining the nature via equivalent or \(O\) Ex 21 Ex 22 Ex 23 Ex 24
    IV) Series-integral comparison and Riemann series
      9) Determining the nature by comparison to an integral --- Riemann series Ex 25 Ex 26 Ex 27 Ex 28
C) Absolute convergence
    I) Definition and main theorem
      10) Showing absolute convergence Ex 29 Ex 30
    II) Comparison by \(O\) for sign-variable series
      11) Reducing a sign-variable series to a positive reference Ex 31 Ex 32 Ex 33
D) Alternating-series criterion
    I) Alternating-series criterion and applications
      12) Applying the alternating-series criterion Ex 34 Ex 35 Ex 36 Ex 37
E) Application\(\virgule\) Stirling's formula
    I) Stirling's formula via the sequence-series link and Wallis
      13) Applying Stirling Ex 38 Ex 39 Ex 40 Ex 41

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