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Summary General_about_numeric_series_license_3_mathematics
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Lecture notes and summaries on numerical series, general information on telescopic series. The document is in English and easy to understand.
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Numerical seriesThey are of the form :
𝜑∶ ℕ⟼ℝ
𝑛 ⟼ 𝜑(𝑛) = 𝑢𝑛
I. Generalities :
1) Definition : Let (𝑢𝑛 ) be a sequence of real numbers (or a numerical series). Let 𝑆𝑛 =
𝑢0 + 𝑢1 + 𝑢2 + ⋯ + 𝑢𝑛 = ∑𝑛𝑘=0 𝑢𝑘 be the sequence of the first 𝑛 + 1 terms of (𝑢𝑛 )𝑛∈ℕ .
i. (𝑆𝑛 )𝑛∈ℕ is called the numerical series with general term 𝑢𝑛 , denoted by : 𝑺𝒏 = ∑+∞
𝟎 𝒖𝒏 .
ii. 𝑆𝑛 = ∑𝑛𝑘=0 𝑢𝑘 is called the 𝑛𝑡ℎ partial sum of the series.
iii. We say that the series ∑𝑛≥0 𝑢𝑛 converges if and only if the sequence of partial sums
(𝑆𝑛 )𝑛∈ℕ converges. In this case, ∃ 𝑙 ∈ ℝ a limit such that :
𝑙 = lim 𝑆𝑛 = lim ∑𝑛𝑘=0 𝑢𝑘 = ∑𝑛≥0 𝑢𝑛 . 𝑙 is called the sum of the numerical series
𝑛→+∞ 𝑛→+∞
∑𝑛≥0 𝑢𝑛 .
iv. A series that does not converge is said to be divergent.
v. The nature of a series with general term 𝑢𝑛 is to determine whether it is convergent or
divergent.
vi. If the sum ∑𝑛≥0 𝑢𝑛 converges to 𝑙, then we have :
𝑛 +∞ 𝑛 +∞ +∞
𝑙 = lim 𝑆𝑛 = lim ∑ 𝑢𝑘 = ∑ 𝑢𝑘 = ∑ 𝑢𝑘 + ∑ 𝑢𝑘 = 𝑆𝑛 + ∑ 𝑢𝑘
𝑛→+∞ 𝑛→+∞
𝑘=0 𝑘=0 𝑘=0 𝑘=𝑛+1 𝑘=𝑛+1
We have then : 𝑙 − 𝑆𝑛 = ∑+∞
𝑘=𝑛+1 𝑢𝑘 = 𝑅𝑛 called the 𝑛𝑡ℎ remainder of the series.
• Proposition : If ∑𝑛≥0 𝑢𝑛 converges, then : lim 𝑅𝑛 = 0.
𝑛→+∞
Let ∑𝑛≥0 𝑢𝑛 be a convergent series with sum 𝑙 : 𝑙 = lim 𝑆𝑛 = lim ∑𝑛𝑘=0 𝑢𝑘 .
𝑛→+∞ 𝑛→+∞
We have : 𝑆𝑛 = 𝑢0 + 𝑢1 + 𝑢2 + ⋯ + 𝑢𝑛−1 + 𝑢𝑛 = 𝑆𝑛−1 + 𝑢𝑛 ⇔ 𝑆𝑛 − 𝑆𝑛−1 = 𝑢𝑛 .
We have : lim 𝑆𝑛 = 𝑙 , and lim 𝑆𝑛−1 = 𝑙 (because 𝑆𝑛−1 is a subsequence of 𝑆𝑛 )
𝑛→+∞ 𝑛→+∞
Then : lim 𝑢𝑛 = lim (𝑆𝑛 − 𝑆𝑛−1 ) = 𝑙 − 𝑙 = 0.
𝑛→+∞ 𝑛→+∞
• Proposition : If ∑𝑛≥0 𝑢𝑛 converges, then lim 𝑢𝑛 = 0.
𝑛→+∞
This is the necessary condition for convergence. Its contrapositive is :
lim 𝑢𝑛 ≠ 0 ⇒ ∑𝑛≥0 𝑢𝑛 diverges.
𝑛→+∞
• Proposition : The geometric series ∑𝑛≥0 𝑟 𝑛 is convergent if |𝑟| < 1 and its sum is
1
𝑙 = 1−𝑟. It is divergent if |𝑟| > 1.
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