........ advanced ... ... ... calculus........ Φ= lim_(n→∞) {∫_1 ^( n) (x/([x]^2 )) dx −ψ(n+1)}=? solution: Φ_n =∫_1 ^( n) (x/([x]^2 )) dx=Σ_(k=1) ^(n−1) ∫_k ^( k+1) (x/k^2 ) dx = (1/2)Σ_(k=1) ^(n−1) (1/k^2 )(2k+1)=Σ_(k=1) ^(n−1) (1/k)+(1/2)Σ_(k=1) ^(n−1) (1/k^2 ) Φ = lim_(n→∞) (Φ_n −ψ(n+1)) = (π^2 /(12)) +lim_(n→∞) (Σ_(k=1) ^(n−1) (1/k)−ψ(n+1)) =_(2 : ψ(n+1)= H_n −γ ) ^(1 :ψ (n+1) := (1/n) +ψ(n)) (π^2 /(12))+lim_(n→∞) (Σ_(k=1) ^(n−1) (1/k)−H_n +γ) ∴ Φ := (π^2 /(12)) +γ −lim_(n→∞) ((1/n)) ......... Φ:=(1/2) ζ (2) +γ γ


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Question Number 139556 by mnjuly1970 last updated on 28/Apr/21
$........ advanced ... ... ... calculus........ Φ= lim_(n→∞) {∫_1 ^( n) (x/([x]^2 )) dx −ψ(n+1)}=? solution: Φ_n =∫_1 ^( n) (x/([x]^2 )) dx=Σ_(k=1) ^(n−1) ∫_k ^( k+1) (x/k^2 ) dx = (1/2)Σ_(k=1) ^(n−1) (1/k^2 )(2k+1)=Σ_(k=1) ^(n−1) (1/k)+(1/2)Σ_(k=1) ^(n−1) (1/k^2 ) Φ = lim_(n→∞) (Φ_n −ψ(n+1)) = (π^2 /(12)) +lim_(n→∞) (Σ_(k=1) ^(n−1) (1/k)−ψ(n+1)) =_(2 : ψ(n+1)= H_n −γ ) ^(1 :ψ (n+1) := (1/n) +ψ(n)) (π^2 /(12))+lim_(n→∞) (Σ_(k=1) ^(n−1) (1/k)−H_n +γ) ∴ Φ := (π^2 /(12)) +γ −lim_(n→∞) ((1/n)) ......... Φ:=(1/2) ζ (2) +γ γ :: Euler− Mascheroni constant...$
$. . . . . . . . a d v a n c e d . . . . . . . . . c a l c u l u s . . . . . . . .$ $Φ = {l i m}_{n \to \infty} {\int_{1}^{n} \frac{x}{{[x]}^{2}} d x - ψ (n + 1)} = ?$ $s o l u t i o n :$ $Φ_{n} = \int_{1}^{n} \frac{x}{{[x]}^{2}} d x = \underset{k = 1}{\sum^{n - 1}} \int_{k}^{k + 1} \frac{x}{k^{2}} d x$ $= \frac{1}{2} \underset{k = 1}{\sum^{n - 1}} \frac{1}{k^{2}} (2 k + 1) = \underset{k = 1}{\sum^{n - 1}} \frac{1}{k} + \frac{1}{2} \underset{k = 1}{\sum^{n - 1}} \frac{1}{k^{2}}$ $Φ = {l i m}_{n \to \infty} (Φ_{n} - ψ (n + 1))$ $= \frac{π^{2}}{12} + {l i m}_{n \to \infty} (\underset{k = 1}{\sum^{n - 1}} \frac{1}{k} - ψ (n + 1))$ $\underset{2 : ψ (n + 1) = H_{n} - γ}{\overset{1 : ψ (n + 1) := \frac{1}{n} + ψ (n)}{=}} \frac{π^{2}}{12} + {l i m}_{n \to \infty} (\underset{k = 1}{\sum^{n - 1}} \frac{1}{k} - H_{n} + γ)$ $∴ Φ := \frac{π^{2}}{12} + γ - {l i m}_{n \to \infty} (\frac{1}{n})$ $. . . . . . . . . Φ := \frac{1}{2} ζ (2) + γ$ $γ :: E u l e r - M a s c h e r o n i c o n s t a n t . . .$
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