Question

The sum of the infinite series $\frac{1.3}{2}+\frac{3.5}{2^2}+\frac{5.7}{2^3}+\frac{7.9}{2^4}+\dots .......\infty .$

Answer 1

$\begin{array}{c}S=\frac{1.3}{2}+\frac{3.5}{2^2}+\frac{5.7}{2^3}+\frac{7.9}{2^4}+.....\\ \frac{S}{2}=\frac{1.3}{2}+\frac{12}{2^2}+\frac{20}{2^3}+\frac{28}{2^4}+.......\\ \frac{S}{2}=\frac{3}{2}+\frac{12}{2^2}+\frac{20}{2^3}+\frac{28}{2^4}+\frac{36}{2^5}+......\\ \\ Let{S}^{\prime }=\frac{12}{2^2}+\frac{20}{2^3}+\frac{28}{2^4}\\ S^{\prime }=\frac{12}{2^2}+\frac{20}{2^3}+\frac{28}{2^4}+\frac{36}{2^5}\\ \frac{S^{\prime }}{2}=\frac{12}{2^3}+\frac{20}{2^4}+\frac{28}{2^5}+.......\\ \frac{S^{\prime }}{2}=\frac{12}{2^2}+\frac{8}{2^3}+\frac{8}{2^4}+\frac{8}{2^5}+.....\\ \frac{S^{\prime }}{2}=3+8\left[\frac{\frac{1}{2^3}}{1-\frac{1}{2}}\right]\\ \frac{S^{\prime }}{2}=2+8\times \frac{1}{8}\times 2\\ \frac{S^{\prime }}{2}=5\\ S^{\prime }=10\\ Now,\\ \frac{S}{2}=\frac{3}{2}+10\\ S=23\end{array}$