mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Blogorrhea > MattcAnderson

Reply
 
Thread Tools
Old 2021-11-26, 14:13   #1
MattcAnderson
 
MattcAnderson's Avatar
 
"Matthew Anderson"
Dec 2010
Oregon, USA

32×131 Posts
Default geometric series

Many of you are familiar with geometric series. Here is a little derivation of a common result.
Finite Geometric Series
Let
S1 = 1 + a + a^2 + ... + a^n.
We multiply S1 by ‘a’ then see
a*S1 = a+ a^2 + … + a^(n+1).
Subtract the second equation from the first one.
(1-a)*S1 = 1-a^(n+1).
Therefore
S1 = [1-a^(n+1)]/(1-a).
We are sure of this. This result about finite geometric series is in many textbooks.
The Wikipedia on this is very good.
The infinite case is another story.
If S2 = 1 + b + b^2 + … is an infinite sum then
S2 converges for -1<b<1.
MattcAnderson is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Prime Numbers as a Function of a Geometric Progression danupp And now for something completely different 2 2017-11-04 17:55
Best 4XX series GPU siegert81 GPU Computing 47 2011-10-14 00:49
Geometric Combinatorial Puzzle davar55 Puzzles 14 2006-04-26 17:27
Another Series Gary Edstrom Puzzles 7 2003-07-03 08:32
Series Rosenfeld Puzzles 2 2003-07-01 17:41

All times are UTC. The time now is 23:34.


Fri Dec 9 23:34:34 UTC 2022 up 113 days, 21:03, 0 users, load averages: 1.10, 0.98, 0.85

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2022, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.

≠ ± ∓ ÷ × · − √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °
∠ ∟ ° ≅ ~ ‖ ⟂ ⫛
≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘ ∏ ∐ ∑ ∧ ∨ ∩ ∪ ⨀ ⊕ ⊗ 𝖕 𝖖 𝖗 ⊲ ⊳
∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟
¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣ … ⋯ ⋮ ⋰ ⋱
∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ
𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎𝜍 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔