Question

Sum of infinity the following series:
$1+\frac{3}{2}+\frac{5}{2^2}+\frac{7}{2^3}+.....$

Answer 1

Let $5=1+\frac{3}{2}+\frac{5}{2^2}+\frac{7}{2^3}+........$

$\Rightarrow \frac{5}{2}=\frac{1}{2}+\frac{3}{2^2}+\frac{5}{2^3}+......$

$(-)$ $(-)$ $(-)$ $(-)$

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$\frac{5}{2}=1+\frac{2}{2}+\frac{2}{2^2}+\frac{1}{2^3}+.......$

$\Rightarrow \frac{5}{2}=2+\frac{1}{2}+\frac{1}{2^2}+......$

$\Rightarrow \frac{5}{2}=2+\frac{\frac{1}{2}}{1-\frac{1}{2}}$

$\Rightarrow 5=2\times 3=6$