EngliSea > M > math > 55 Analysis > 4 Series
 
4 Series
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411 Geometric Series
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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 ...
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Proverbs and Quotes in English
Short sentences and easy words.
 
415 p-series
The p-series 「1/n˄p Σn=1,∞」 converges if p>1 and diverges if p≤1.
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415 Harmonic series
「1/n Σn=1,∞」 diverges
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415 「1/n² Σn=1,∞」
「1/n² Σn=1,∞」 converges
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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⸥┆ ...
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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
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A first look at series
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Power Series
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◌◌◌ 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.
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