Expert: Josh - 8/23/2004

Question
I have one question for you, kindly help me.

1. The infinite sum 1 + 1/5 + 4/5^2 + 9/5^3 + 16/5^4 ... equals to what?


Answer
Abir,

The infinite sum 1 + 1/5 + 4/5^2 + 9/5^3 + 16/5^4 ......can be written as 1 + SUM_i (i^2)/(5^i).
So, let S = SUM (i^2)/(5^i) from i=1..infinity.

Now, let g(a)=SUM 1/(5^i) from i=a to i=infinity,
g(a)=1/[4*5^(a-1)].

Writing this sum incrementally,
S = 1*[1/5 + 1/5^2 + 1/5^3 + 1/5^4 +...]
    +(4-1)*[1/5^2 + 1/5^3 + 1/5^4 + 1/5^5 +...]
    +(9-4)*[1/5^3 + 1/5^4 + 1/5^5 + 1/5^6 +...]
    +(16-9)*[1/5^4 + 1/5^5 + 1/5^6 + 1/5^7 +...]
    +...
   = 1*g(1)+(2^2-1)*g(2)+(3^2-2^2)*g(3)+(4^2-3^2)*g(4)+...
   = (1/4) * SUM (2*i-1)/(5^(i-1)), where i=1..infinity
   = (1/4) * (15/8)
   = 15/32

Thus, the original sum = 1+15/32.

This is quite hard to come up with, especially not knowing where you are at in class and what your teacher has taught in previous lessons.

Josh