What is the Convergent Series Sum Formula?

[jediwhelan]

Homework Statement

Dear All,

I have a series that I know to converge but for which I can't work out the infinite sum. It should be something simple.

$$S_n = \sum_{j=1}^\infty \beta^j j$$

Can somebody help me with this?

I think the solution is:

$$\frac{\beta}{(1-\beta)^2}$$

 

[Gregg]

It looks right:

$$S_n = \beta + 2\beta^2 + 3\beta^3 + \cdots$$

$$\frac{S_n}{\beta} = 1 + 2\beta + 3\beta^2 + \cdots$$

$$(\frac{1}{\beta}-1) S_n = 1 + \beta + \beta^2 + \cdots$$

$$(\frac{1}{\beta}-1) S_n = \frac{1}{1-\beta}$$

$$(\frac{1-\beta}{\beta}) S_n = \frac{1}{1-\beta}$$

$$S_n = \frac{\beta}{(1-\beta)^2}$$

It's called an arithmetico-geometric series I think,

$$\displaystyle\sum_{n=0}^{\infty}(a+nd)r^n = \frac{a}{1-r} + \frac{rd}{(1-r)^2}$$

 

[jediwhelan]

brilliant. I was trying something like that but couldn't get it.

Thanks for the quick reply.

Paul