Question:
The sum of the infinite series
(1/5)*( (1/5) − (1/7) )
+ (1/5)^2*( (1/5)^2 − (1/7)^2 )
+ (1/5)^3*( (1/5)^3 − (1/7)^3 )
+ ...... 
is equal to ?

Solution:

Let a = 1/5 and b = 1/7. The series is
S = Σ_{n=1 to ∞} a^n( a^n − b^n )
  = Σ_{n=1 to ∞} ( a^{2n} − (ab)^n )

Separate the sums (both geometric, |a|<1, |ab|<1):
S = Σ_{n=1 to ∞} a^{2n}  −  Σ_{n=1 to ∞} (ab)^n
  = (a^2)/(1 − a^2)  −  (ab)/(1 − ab)

Substitute a = 1/5, b = 1/7:
a^2 = 1/25,   ab = 1/35

S = (1/25)/(1 − 1/25)  −  (1/35)/(1 − 1/35)
  = (1/25)/(24/25)  −  (1/35)/(34/35)
  = 1/24  −  1/34
  = (34 − 24) / (24·34)
  = 10 / 816
  = 5 / 408

Answer:
S = 5/408