Notes and course summaries on harmonic series, all the rules and theorems you need to solve your exercises. The document is in English that is easy to read and understand.
3) Harmonic series :
𝟏
They are of the form : ∑𝒏≥𝟏 𝒏
• Reminders :
1. A numerical sequence of real numbers (𝑎𝑛 )𝑛 is a Cauchy sequence if : ∀𝜀 > 0 ; ∃𝑁 ∈ ℕ ,
such that : ∀𝑝 > 𝑞 > 𝑁, we have : |𝑎𝑝 − 𝑎𝑞 | < 𝜀.
2. (𝑎𝑛 )𝑛 is a convergente sequence if : ∃𝑙 ∈ ℝ , such that : lim 𝑎𝑛 = 𝑙 ⟺ ∀𝜀 > 0 , ∃𝑛0 ; such
𝑛→+∞
that : ∀𝑛 > 𝑛0 , we have : |𝑢𝑛 − 𝑙| < 𝜀.
3. The real number sequence (𝑎𝑛 )𝑛 is a Cauchy sequence if and only if it is convergent.
4. (𝑎𝑛 )𝑛 is a sequence that is not Cauchy if : ∃𝜀0 > 0 ; ∀𝑁 ∈ ℕ ; ∃𝑝0 , 𝑞0 > 𝑁, such that :
|𝑎𝑝0 − 𝑎𝑞0 | ≥ 𝜀0 .
1
Let's consider the sequence of partial sums 𝑆𝑛 = ∑𝑛𝑘=1 𝑘 , and let 𝑝0 = 2𝑛 , 𝑞0 = 𝑛.
1 1
We have : 𝑆𝑝0 − 𝑆𝑞0 = 𝑆2𝑛 − 𝑆𝑛 = ∑2𝑛 𝑛
𝑘=1 𝑘 − ∑𝑘=1 𝑘
𝑛 1
𝑆2𝑛 − 𝑆𝑛 ≥ =
2𝑛 2
1
We have the following : ∃𝜀0 = 2 ; ∀𝑛 ; ∃𝑝0 = 2𝑛 ; ∃𝑞0 = 𝑛 , such that :
1
|𝑆2𝑛 − 𝑆𝑛 | > 𝜀0 =
2
1
Then : (𝑆𝑛 )𝑛 diverges, so the harmonic series ∑𝑛≥1 𝑛 diverges.
• Properties and operations on series :
1. The nature of a series is not changed if we add or subtract a finite number of its terms.
2. In the case of convergence, it is the sum that changes when adding or subtracting terms. The
series ∑𝑛≥0 𝑢𝑛 and ∑𝑛≥𝑛0 𝑢𝑛 remains of the same nature but with a different sum.
1
Example : ∑𝑛≥0 (𝑛+1)(𝑛+2) = 1
1 1 1 1 1 1
Let be : ∑𝑛≥2 (𝑛+1)(𝑛+2) ; we have : 𝑇𝑛 = ∑𝑛𝑘=2 (𝑘+1)(𝑘+2) = ∑𝑛𝑘=2 (𝑘+1 − 𝑘+2) = 3 − 𝑛+2
1 1 1
lim 𝑇𝑛 = 3 ; so : ∑𝑛≥2 (𝑛+1)(𝑛+2) = 3
𝑛→+∞
3. Let ∑𝑛≥0 𝑢𝑛 and ∑𝑛≥0 𝑣𝑛 be two series. If ∑𝑛≥0 𝑢𝑛 converges with sum 𝑆 and ∑𝑛≥0 𝑣𝑛
converges with sum 𝑆′, then the series ∑𝑛≥0(𝑢𝑛 + 𝑣𝑛 ) converges with sum 𝑆 + 𝑆′.
4. Let 𝛼 ∈ ℝ∗ . If ∑𝑛≥0 𝑢𝑛 converges with sum 𝑆, then the series ∑𝑛≥0(𝛼𝑢𝑛 ) converges with sum
𝛼𝑆.
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