calculer-la-somme-n-1-oo-1-n-n-2-x-n-

Question Number 166291 by SANOGO last updated on 17/Feb/22

c a l c u l e r l a s o m m e

\underset{n = 1}{\sum^{+ o o}} \frac{1}{n (n + 2)} x^{n}

Answered by TheSupreme last updated on 18/Feb/22

Σ = a_{n} x^{n}

ρ = l i \underset{n}{m} \frac{a_{n}}{a_{n + 1}} = 1

∣ x ∣⩽ 1

Σ = \frac{1}{2} \sum_{n} (\frac{1}{n} - \frac{1}{n + 2}) x^{n}

Σ = \frac{1}{2} Σ \frac{x^{n}}{n} - \frac{1}{2} Σ \frac{x^{n}}{n + 2}

Σ \frac{x^{n}}{n} = Σ \int x^{n - 1} d x = \int Σ x^{n - 1} = \int \frac{1}{1 - x} d x = - l n ∣ 1 - x ∣

Σ \frac{x^{n}}{n + 2} = \frac{1}{x^{2}} Σ \frac{x^{n + 2}}{n + 2} = Σ \int x^{n + 1} d x = \int Σ x^{n + 1} d x = \int (\frac{1}{1 - x} - 1 - x) d x

- l n ∣ 1 - x ∣ - x - \frac{x^{2}}{2}

Σ \frac{x^{n}}{n (n + 2)} = \frac{1}{2} (\frac{x^{2}}{2} + x)

Commented by SANOGO last updated on 18/Feb/22

m e r c i b i e n

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