Ratio test (convergent or divergent?)

[freshcoast]

Homework Statement

∞
Ʃ n / 2^n
n=1

Homework Equations

ratio test
lim |a(n+1) / a(n)|
n->∞

The Attempt at a Solution

I have the answer and the steps its just there's one part I am confused on,
first I just apply n+1 to all my n terms, which gives me,

∞
Ʃ [(n+1)/2^(n+1)] / [n/2^n]
n=1

and if I multiply the a(n+1) part by the reciprocal of an, I don't understand how the terms 2^(n+1) and 2^n cancel each other out and leaves me with just a 2?

 

[member 428835]

consider rules of exponents (which boils down to multiplication). consider $$x*x=x^2$$ this is the same as $$x^1*x^1=x^{1+1}=x^2$$ thus it seems plausible that $$x^{n+1}=x^n*x^1$$ does this answer your question?

 

[Mark44]

Homework Statement

∞
Ʃ n / 2^n
n=1

Homework Equations

ratio test
lim |a(n+1) / a(n)|
n->∞

The Attempt at a Solution

I have the answer and the steps its just there's one part I am confused on,
first I just apply n+1 to all my n terms, which gives me,

∞
Ʃ [(n+1)/2^(n+1)] / [n/2^n]
n=1

No it doesn't. In the Ratio Test you look at the ratio a_n+1/a_n, NOT the sum of that ratio.

and if I multiply the a(n+1) part by the reciprocal of an, I don't understand how the terms 2^(n+1) and 2^n cancel each other out and leaves me with just a 2?