find-S-1-1-1-1-1-1-1-1-1-1-1-1-2-1-2-1-2-1-3-1-3-1-3-1-4-1-4-1-4-1-5-

Question Number 206393 by MaruMaru last updated on 13/Apr/24

find S=1+Σ_ℓ (((−)^ℓ )/ℓ)((1/ℓ)−(1/(ℓ+1))) , ℓ∈[1,∞) 1+Σ_ℓ (((−)^ℓ )/ℓ)((1/ℓ)−(1/(ℓ+1))) 1−(1−(1/2))+(1/2)((1/2)−(1/3))−(1/3)((1/3)−(1/4))+(1/4)((1/4)−(1/5))−......

$$\mathrm{find}\:\mathrm{S}=\mathrm{1}+\underset{\ell} {\sum}\:\frac{\left(−\right)^{\ell} }{\ell}\left(\frac{\mathrm{1}}{\ell}−\frac{\mathrm{1}}{\ell+\mathrm{1}}\right)\:,\:\ell\in\left[\mathrm{1},\infty\right) \\ $$$$\mathrm{1}+\underset{\ell} {\sum}\:\frac{\left(−\right)^{\ell} }{\ell}\left(\frac{\mathrm{1}}{\ell}−\frac{\mathrm{1}}{\ell+\mathrm{1}}\right) \\ $$$$\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right)+\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}}\right)−\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{4}}\right)+\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{5}}\right)−…… \\ $$

Answered by MaruMaru last updated on 17/Apr/24

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