The sum of the infinite series $$\cfrac{1}{5}\left(\cfrac{1}{5} - \cfrac{1}{7}\right) + \left(\cfrac{1}{5}\rig

Question

The sum of the infinite series $$\cfrac{1}{5}\left(\cfrac{1}{5} - \cfrac{1}{7}\right) + \left(\cfrac{1}{5}\right)^2 \left(\left(\cfrac{1}{5}\right)^2 - \left(\cfrac{1}{7}\right)^2\right) + \left(\cfrac{1}{5}\right)^3 \left(\left(\cfrac{1}{5}\right)^3 - \left(\cfrac{1}{7}\right)^3\right) + ......$$ is equal to

Accepted Answer

5 408 \cfrac{5}{408} 408 5 ​. Opening the brackets, we get the series as: ( 1 5 ) 2 − ( 1 5 × 1 7 ) + ( 1 5 ) 4 − ( 1 5 × 1 7 ) 2 + ( 1 5 ) 6 − ( 1 5 × 1 7 ) 6 + . . . \left(\dfrac{1}{5}ight)^2-\left(\dfrac{1}{5}	imes\ \dfrac{1}{7}ight)+\left(\dfrac{1}{5}ight)^4-\left(\dfrac{1}{5}	imes\ \dfrac{1}{7}ight)^2+\left(\dfrac