Pillow English Listening

• 4 Series
9730#9737 SIBLINGS CHILDREN 9737

• 411 Geometric Series
9730#9729 SIBLINGS CHILDREN 9729

• 411 Geometric Series
ᐥThe geometric series ｢a·x˄r Σr=0,∞｣ = a + a·x + a·x² + ... (a≠0) converges if and only if |x|<1. Moreover, its sum is then a/(1−r).ᐥ Partial sum, (1−x)·｢a·x˄r Σr=0,n−1｣ = ｢a·x˄r Σr=0,n−1｣ − ｢a·x˄(r+1) Σr=0,n−1｣ = ｢a·x˄r Σr=0,n−1｣ ┆｢a·x˄(r+1) Σr=0,n−1 ...
9730#9731 SIBLINGS CHILDREN COMMENT 9731

• 415 p-series
The p-series ｢1/n˄p Σn=1,∞｣ converges if p>1 and diverges if p≤1.
9730#473 SIBLINGS CHILDREN 473

• 415 Harmonic series
｢1/n Σn=1,∞｣ diverges
9730#3652 SIBLINGS CHILDREN 3652

• 415 ｢1/n² Σn=1,∞｣
｢1/n² Σn=1,∞｣ converges
9730#24853 SIBLINGS CHILDREN 24853

• 421 First comparison test
ᐥIf 0 ≤ a⸤n⸥ ≤ b⸤n⸥ for all n∈ℕ then (a) ｢b⸤i⸥ Σi=1,∞｣ converges ⇒ ｢a⸤i⸥ Σi=1,∞｣ converges (b) ｢a⸤i⸥ Σi=1,∞｣ diverges ⇒ ｢b⸤i⸥ Σi=1,∞｣ divergesᐥ Proof of (a) ┆0 ≤ a⸤n⸥┆ ｢a⸤i⸥ Σi=1,n｣ is increasing. ｢a⸤i⸥ Σi=1,n｣ ┆a⸤n⸥ ≤ b⸤n⸥┆ ≤ ｢b⸤i⸥ Σi=1,n｣ ┆0 ≤ b⸤n⸥┆ ...
9730#5038 SIBLINGS CHILDREN 5038

• 422 Second comparison test
ᐥLet ｢a⸤n⸥ Σn=1,∞｣ and ｢b⸤n⸥ Σn=1,∞｣ be positive-term series such that ｢a⸤n⸥/b⸤n⸥ Σn=1,∞｣ = L ≠ 0. Then ｢a⸤n⸥ Σn=1,∞｣ converges if and only if ｢b⸤n⸥ Σn=1,∞｣ converges.ᐥ
9730#9785 SIBLINGS CHILDREN 9785

• A first look at series
9730#9912 SIBLINGS CHILDREN 9912

• Power Series
9730#9780 SIBLINGS CHILDREN 9780

◌◌◌ Series
A series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.
9730#5170 SIBLINGS CHILDREN 5170