trustle > EngliSea > Corona Safe Schooling > mathematics > Quick Math Index > 55 Analysis > 4 Series
 
4 Series
9737
 
411 Geometric Series
9729
 
411 Geometric Series
The 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).
9731 COMMENT
 
415 p-series
The p-series 「1/n˄p Σn=1,∞」 converges if p>1 and diverges if p≤1.
473
 
415 Harmonic series
「1/n Σn=1,∞」 diverges
3652
 
415 「1/n² Σn=1,∞」
「1/n² Σn=1,∞」 converges
511
 
421 First comparison test
If 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,∞」 diverges
5038
 
422 Second comparison test
Let 「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.
9785
 
A first look at series
9912
 
Power Series
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.
5170

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