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Question Number 211421 by universe last updated on 08/Sep/24

if a_n = n^4 ∫_n ^(n+1) ((x dx)/(1+x^5 )) then (1) Σa_n is convergent or divergent?? (2) lim_(n→∞) a_(n ) = ??

$$\:\:\:\:\mathrm{if}\:\mathrm{a}_{\mathrm{n}} \:=\:\mathrm{n}^{\mathrm{4}} \int_{\mathrm{n}} ^{\mathrm{n}+\mathrm{1}} \:\frac{\mathrm{x}\:\mathrm{dx}}{\mathrm{1}+\mathrm{x}^{\mathrm{5}} }\:\:\mathrm{then} \\ $$$$\:\:\:\:\left(\mathrm{1}\right)\:\Sigma\mathrm{a}_{\mathrm{n}} \:\mathrm{is}\:\mathrm{convergent}\:\mathrm{or}\:\mathrm{divergent}?? \\ $$$$\:\:\:\:\left(\mathrm{2}\right)\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{a}_{\mathrm{n}\:} \:=\:?? \\ $$

Answered by mathmax last updated on 09/Sep/24

n≤x≤n+1 et 1+n^5 ≤1+x^5 ≤1+(n+1)^5 ⇒ (1/(1+(n+1)^5 ))≤(1/(1+x^5 ))≤(1/(1+n^5 )) ⇒((n^4 ×n)/((n+1)^5 +1))≤n^4 (x/(1+x^5 ))≤((n^4 (n+1))/(1+n^5 )) ⇒(n^5 /((n+1)^5 +1))≤n^4 ∫_n ^(n+1) ((xdx)/(1+x^5 ))≤((n^5 +n)/(n^5 +1)) ⇒ lim_(n→+∞) a_n =1 et Σa_n est div.

$${n}\leqslant{x}\leqslant{n}+\mathrm{1}\:\:\:\:{et}\:\mathrm{1}+{n}^{\mathrm{5}} \leqslant\mathrm{1}+{x}^{\mathrm{5}} \leqslant\mathrm{1}+\left({n}+\mathrm{1}\right)^{\mathrm{5}} \:\Rightarrow \\ $$$$\frac{\mathrm{1}}{\mathrm{1}+\left({n}+\mathrm{1}\right)^{\mathrm{5}} }\leqslant\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{5}} }\leqslant\frac{\mathrm{1}}{\mathrm{1}+{n}^{\mathrm{5}} }\:\Rightarrow\frac{{n}^{\mathrm{4}} ×{n}}{\left({n}+\mathrm{1}\right)^{\mathrm{5}} +\mathrm{1}}\leqslant{n}^{\mathrm{4}} \frac{{x}}{\mathrm{1}+{x}^{\mathrm{5}} }\leqslant\frac{{n}^{\mathrm{4}} \left({n}+\mathrm{1}\right)}{\mathrm{1}+{n}^{\mathrm{5}} } \\ $$$$\Rightarrow\frac{{n}^{\mathrm{5}} }{\left({n}+\mathrm{1}\right)^{\mathrm{5}} +\mathrm{1}}\leqslant{n}^{\mathrm{4}} \int_{{n}} ^{{n}+\mathrm{1}} \frac{{xdx}}{\mathrm{1}+{x}^{\mathrm{5}} }\leqslant\frac{{n}^{\mathrm{5}} +{n}}{{n}^{\mathrm{5}} +\mathrm{1}}\:\Rightarrow \\ $$$${lim}_{{n}\rightarrow+\infty} {a}_{{n}} =\mathrm{1}\:\:\:{et}\:\Sigma{a}_{{n}} \:{est}\:{div}. \\ $$

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