Series Convergence

Question

I'm having hard time understanding how following series converges ; 

1 + 2/5 + 3/5^2 + 4/5^3 + 5/5^4 + ............ infinity

Comments

the denominator is increasing as a much faster rate, so eventually it should become 0.

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I've found this could be reduced to {1-1/5}^-2 ; which is binomial series.I was so fixated on this being two series u_n & v_n.That made things quite complicated !

Answer 1

S      = 1 + 2/5 + 3/5^2 + 4/5^3 + 5/5^4 + ............ infinity

S/5  =       1/5 + 2/5^2 + 3/5^3 + 4/5^4 + 5/5^5 + ............ infinity

S-S/5 = 1 + 1/5 + 1/5^2 + 1/5^3 ........... infinity

4S/5 = 1/(1-1/5)= 5/4

S= 25/16

Comments

I found this really intuitive , thanks.