The sum of the infinite series `1+(1+1/2)(1/3)+(1+1/2+1/(2^2))(1/(3^2))+ ....oo`

The sum of the infinite series 1+(1+1/5)(1/2)+(1+1/5+1/(5^2))(1/(2^2))+...

The sum of the infinite series 1+(1+1/5)(1/2)+(1+1/5+1/(5^2))(1/(2^2))+...

The sum of the infinite series (1)/(2) ((1)/(3) + (1)/(4)) - (1)/(4)((1)/(3^(2)) + (1)/(4^(2))) + (1)/(6) ((1)/(3^(3)) + (1)/(4^(3))) - ...is equal to

The sum of the infinite series 1+(1+(1)/(5))((1)/(2))+(1+(1)/(5)+(1)/(5^(2)))((1)/(2^(2)))+