series comparison test?

[sambellamy]

The book problem gives: 0.d₁ d₂ d₃ d₄ = d1/10 + d2/10^2 + d3/10^3, etc. They ask to show that the series always converges.
I understand that this is probably a geometric series. i have a=d and r=1/10^d, or 10^-d. I do not know where to go from here, or what to compare it to. I am pretty sure I am supposed to use a comparison test. should I rewrite it as (d)(1/10)(1/10)^d-1? Thanks in advance for help!

 

[Ishuda]

Just to be complete, you will need another piece of information and that is that the d_j are ('almost') bounded. That is, in the simplest sense, there is some positive D belonging to the reals such that
|d_j| < D for all j
That being the case,
$\displaystyle \frac{d_j}{10^j} < \frac{D}{10^j}$
and your conjecture about using the comparison test is justified.

The 'almost' means that the d_j don't have to be strictly bounded, they just have to be bounded in such a way that $\displaystyle \frac{d_j}{10^j}$ goes to zero fast enough. That is, something like:
There exists positive real numbers x and D such that 0 < x < 10 and
$\displaystyle |d_j| < D\space x^j$ for all j.

Edit to add: And even that last is too restrictive.