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Nice-Calculus-Evaluate-n-1-1-




Question Number 138823 by mnjuly1970 last updated on 18/Apr/21
                 ......Nice  ... Calculus......         Evaluate::     𝛗=Σ_(n=1) ^∞ (1/)=?
$$\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:……\mathscr{N}{ice}\:\:…\:\mathscr{C}{alculus}…… \\ $$$$\:\:\:\:\:\:\:\mathscr{E}{valuate}::\:\:\:\:\:\boldsymbol{\phi}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{}=? \\ $$
Answered by qaz last updated on 18/Apr/21
φ=Σ_(n=1) ^∞ (1/ (((n+2)),(3) ))=Σ_(n=1) ^∞ (6/((n+2)(n+1)n))  =6∫_0 ^1 dx∫_0 ^x dx∫_0 ^x Σ_(n=1) ^∞ x^(n−1) dx  =6∫_0 ^1 dx∫_0 ^x dx∫_0 ^x (dx/(1−x))  =−6∫_0 ^1 dx∫_0 ^x ln(1−x)dx  =6∫_0 ^1 xlnx+1−xdx  =(9/2)
$$\phi=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\begin{pmatrix}{{n}+\mathrm{2}}\\{\mathrm{3}}\end{pmatrix}}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{6}}{\left({n}+\mathrm{2}\right)\left({n}+\mathrm{1}\right){n}} \\ $$$$=\mathrm{6}\int_{\mathrm{0}} ^{\mathrm{1}} {dx}\int_{\mathrm{0}} ^{{x}} {dx}\int_{\mathrm{0}} ^{{x}} \underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{x}^{{n}−\mathrm{1}} {dx} \\ $$$$=\mathrm{6}\int_{\mathrm{0}} ^{\mathrm{1}} {dx}\int_{\mathrm{0}} ^{{x}} {dx}\int_{\mathrm{0}} ^{{x}} \frac{{dx}}{\mathrm{1}−{x}} \\ $$$$=−\mathrm{6}\int_{\mathrm{0}} ^{\mathrm{1}} {dx}\int_{\mathrm{0}} ^{{x}} {ln}\left(\mathrm{1}−{x}\right){dx} \\ $$$$=\mathrm{6}\int_{\mathrm{0}} ^{\mathrm{1}} {xlnx}+\mathrm{1}−{xdx} \\ $$$$=\frac{\mathrm{9}}{\mathrm{2}} \\ $$
Commented by mnjuly1970 last updated on 18/Apr/21
grateful...
$${grateful}… \\ $$

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