Ответ :

[tex]\sum\limits _{n-1}^{\infty }\dfrac{\sqrt{n+1}}{(4n-3)\cdot 5^{n}}=\dfrac{\sqrt2}{1\cdot 5^2}+\dfrac{\sqrt3}{9\cdot 5^3}+\dfrac{\sqrt4}{13\cdot 5^4}+...=\dfrac{\sqrt2}{25}+\dfrac{\sqrt3}{1125}+\dfrac{2}{40625}+...[/tex]