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p-n-nth-prime-p-1-2-p-2-3-p-3-5-Do-the-following-sums-converge-Prove-disprove-1-S-n-1-n-p-n-2-S-n-1-n-p-n-2-




Question Number 7723 by FilupSmith last updated on 12/Sep/16
p_n =nth prime  (p_1 =2,  p_2 =3,   p_3 =5, ...)  Do the following sums converge? Prove/disprove.  (1)        S=Σ_(n=1) ^∞ (n/p_n )  (2)        S=Σ_(n=1) ^∞ (n/p_n ^2 )
$${p}_{{n}} ={n}\mathrm{th}\:\mathrm{prime}\:\:\left({p}_{\mathrm{1}} =\mathrm{2},\:\:{p}_{\mathrm{2}} =\mathrm{3},\:\:\:{p}_{\mathrm{3}} =\mathrm{5},\:…\right) \\ $$$$\mathrm{Do}\:\mathrm{the}\:\mathrm{following}\:\mathrm{sums}\:\mathrm{converge}?\:\mathrm{Prove}/\mathrm{disprove}. \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:{S}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}}{{p}_{{n}} } \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:{S}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}}{{p}_{{n}} ^{\mathrm{2}} } \\ $$
Commented by FilupSmith last updated on 12/Sep/16
is the following correct for (1):  lim_(n→∞) (n/p_n )=((lim_(n→∞) n)/(lim_(n→∞) p_n ))=(∞/∞)  L′hopital  (((d/dn)(n))/((d/dn)(p_n )))  =((d/dn)(p_n ))^(−1)   =???
$$\mathrm{is}\:\mathrm{the}\:\mathrm{following}\:\mathrm{correct}\:\mathrm{for}\:\left(\mathrm{1}\right): \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{n}}{{p}_{{n}} }=\frac{\underset{{n}\rightarrow\infty} {\mathrm{lim}}{n}}{\underset{{n}\rightarrow\infty} {\mathrm{lim}}{p}_{{n}} }=\frac{\infty}{\infty} \\ $$$$\mathrm{L}'\mathrm{hopital} \\ $$$$\frac{\frac{{d}}{{dn}}\left({n}\right)}{\frac{{d}}{{dn}}\left({p}_{{n}} \right)} \\ $$$$=\left(\frac{{d}}{{dn}}\left({p}_{{n}} \right)\right)^{−\mathrm{1}} \\ $$$$=??? \\ $$

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