series question (1 Viewer)

[blackops23]

-1<=x<=1

observe that:

1=1
2x=x+x
3x^2=x^2 + x^2 + x^2
4x^3=x^3 + x^3 + x^3 + x^3

by studying the above arrangements or otherwise find in simplest algebraic form an expression for the limiting sum of the series:

1 + 2x + 3x^2 + 4x^3 + ... + nx^[n -1]

 

[Sy123]

-1<=x<=1

observe that:

1=1
2x=x+x
3x^2=x^2 + x^2 + x^2
4x^3=x^3 + x^3 + x^3 + x^3

by studying the above arrangements or otherwise find in simplest algebraic form an expression for the limiting sum of the series:

1 + 2x + 3x^2 + 4x^3 + ... + nx^[n -1]

put the terms like a right angle triangle and add the columns:

1
x+x
x^2+x^2
x^3+x^3+x^3
x^4+x^4+x^4+x^4
.
.
.

 

[RealiseNothing]

 

[blackops23]

thanks

 

[RealiseNothing]

Alternatively consider the series:



Differentiate both sides: