Question Number 151841 by mathdanisur last updated on 23/Aug/21
$$\mathrm{The}\:\mathrm{volue}\:\mathrm{of}\:\mathrm{the}\:\mathrm{limit}:\: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{2}^{−\boldsymbol{\mathrm{n}}^{\mathrm{2}} } }{\underset{\boldsymbol{\mathrm{k}}=\boldsymbol{\mathrm{n}}+\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{2}^{−\boldsymbol{\mathrm{k}}^{\mathrm{2}} } }\:\:\:;\:\:\:\left(\mathrm{a}\right)\mathrm{0}\:\:\left(\mathrm{b}\right)\mathrm{some}\:\mathrm{c}\in\left(\mathrm{0};\mathrm{1}\right)\:\:\left(\mathrm{c}\right)\mathrm{1} \\ $$