Let-consider-an-integer-serie-a-n-x-n-given-by-a-n-H-n-k-1-n-1-k-1-Find-out-the-largest-domain-D-of-convergence-of-that-integer-serie-2-x-D-explicit-the-sum-S-x-of-the-a-n-x-n

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Question Number 66814 by ~ À ® @ 237 ~ last updated on 20/Aug/19

Let consider an integer serie {a_n x^n } given by a_n = H_n =Σ_(k=1) ^n (1/k) 1) Find out the largest domain D of convergence of that integer serie 2) ∀ x∈D , explicit the sum S(x) of the {a_n x^n } 3) Calculate ∫_(−1) ^1 S(1−x)S(x) dx .

$${Let}\:{consider}\:{an}\:{integer}\:{serie}\:\left\{{a}_{{n}} {x}^{{n}} \right\}\:{given}\:{by}\:\:{a}_{{n}} \:=\:{H}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}\: \\ $$$$\left.\mathrm{1}\right)\:{Find}\:{out}\:{the}\:{largest}\:{domain}\:{D}\:{of}\:{convergence}\:{of}\:{that}\:{integer}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:\forall\:{x}\in{D}\:\:,\:{explicit}\:{the}\:{sum}\:{S}\left({x}\right)\:{of}\:{the}\:\left\{{a}_{{n}} {x}^{{n}} \right\}\: \\ $$$$\left.\mathrm{3}\right)\:{Calculate}\:\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:{S}\left(\mathrm{1}−{x}\right){S}\left({x}\right)\:{dx}\:. \\ $$$$ \\ $$$$\:\:\:\: \\ $$

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Let-consider-an-integer-serie-a-n-x-n-given-by-a-n-H-n-k-1-n-1-k-1-Find-out-the-largest-domain-D-of-convergence-of-that-integer-serie-2-x-D-explicit-the-sum-S-x-of-the-a-n-x-n

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