Series (mathematics)

Series - Definition

Let (𝑎_𝑘)_𝑘∊ℕ be a sequence. The sequence (𝑠_𝑛)_𝑛∊ℕ given by:

• 𝑠_𝑛 = 𝛴_{1≤𝑘≤𝑛}𝑎_𝑘

is called a series.

If (𝑠_𝑛)_𝑛∊ℕ is convergent, we write:

• 𝛴_{1≤𝑘≤∞}𝑎_𝑘 = 𝑙𝑖𝑚_𝑛→∞𝑠_𝑛 = 𝑙𝑖𝑚_𝑛→∞𝛴_{1≤𝑘≤𝑛}𝑎_𝑘

Series - Properties

If 𝛴_{1≤𝑘≤𝑛}𝑎_𝑘and 𝛴_{1≤𝑘≤𝑛}𝑏_𝑘 are both convergent, then:

• 𝛴_{1≤𝑘≤𝑛}(𝑎_𝑘 + 𝑏_𝑘) is also convergent with the limit 𝛴_{1≤𝑘≤𝑛}(𝑎_𝑘 + 𝑏_𝑘) = 𝛴_{1≤𝑘≤𝑛}(𝑎_𝑘) + 𝛴_{1≤𝑘≤𝑛}(𝑏_𝑘)
• 𝛴_{1≤𝑘≤𝑛}(𝜆·𝑎_𝑘) is also convergent with the limit 𝛴_{1≤𝑘≤𝑛}(𝜆·𝑎_𝑘) = 𝜆·𝛴_{1≤𝑘≤𝑛}(𝑎_𝑘)

Example Series

• Succession
• Harmonic Series