2014/046) solve for a :a + 2a^2 + 3a^3 + ... = 30

Because the series converges so we must have

|a| < 1

Now we are given
a + 2a^2 + 3a^3 + ... = 30 ... (1)

 Multiply (1) by a to get

a^2 +  2a^3 + 3a^4 + ... = 30a ... (2)

 Subtract (2) from (1) to get

a + a^2 + a^3 + ... = 30(1-a)  ... (1)

 Or a/(1-a) = 30(1-a)

Or 30(1-a)^2 = a

Or(30 – 61 a + 30a^2) = 0

Or (5-6a)(6-5a) = 0

hence a = 5/6 o 6/5

As |a| < 1 so a = 5/6