Question

Evaluate 3 x 31/2 x 31/4 x 31/8 x ... to infinity​

Answer 1

SOLUTION

TO DETERMINE

$\displaystyle \sf{3 \times {3}^{ \frac{1}{2} } \times {3}^{ \frac{1}{4} } \times {3}^{ \frac{1}{8} } \times .. \: infinity }$

FORMULA TO BE IMPLEMENTED

If in a Geometric Progression with first a and common ratio = r ( < 1 ) then sum of infinite number of terms

$\displaystyle \sf{ = \frac{a}{1 - r} }$

EVALUATION

$\displaystyle \sf{3 \times {3}^{ \frac{1}{2} } \times {3}^{ \frac{1}{4} } \times {3}^{ \frac{1}{8} } \times .. \: infinity }$

$\displaystyle \sf{ = {3}^{1} \times {3}^{ \frac{1}{2} } \times {3}^{ \frac{1}{4} } \times {3}^{ \frac{1}{8} } \times .. \: infinity }$

$\displaystyle \sf{ = {3}^{1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + .. \: infinity } }$

$\displaystyle \sf{ = {3}^{ \frac{1}{1 - \frac{1}{2} } } }$

$\displaystyle \sf{ = {3}^{ \frac{1}{ \frac{1}{2} } } }$

$\displaystyle \sf{ = {3}^{2} }$

$= 9$

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