Question: Show $b\_0$,.....,$b\_{n-1}$ are real numbers, then for $k$ sufficiently large,

$\frac{1}{2} < 1 + \frac{b\_{n-1}}{k} +$...$+\frac{b\_0}{k^n} < \frac{3}{2}$. 

Right now I am studying Tests of Converges of Series, so my partial solution tries this approach. please help me with my solution.

**solution:**

$\frac{1}{2} < 1 + \frac{b\_{n-1}}{k} +$...$+\frac{b\_0}{k^n} < \frac{3}{2}$. 

$\frac{k^n}{2} < k^n + b\_{n-1}k^{n-1} +$...$+b\_0< \frac{3k^n}{2}$. 

$\frac{k^n}{2} < k^n(1 + b\_{n-1}(\frac{1}{k}) +$...$+b\_0({\frac{1}{k^n}})) < \frac{3k^n}{2}$. 

Let $c\_k =1 + b\_{n-1}(\frac{1}{k}) +$...$+b\_0({\frac{1}{k^n}})$

$c\_k \leq \frac{3k^n/2}{k^n/2} = 3$

Let $a\_k = k^nc\_k \leq 3k^n$ for sufficiently large $k$.

Answer: 
Generalizing
André Nicolas's
answer:

As he showed,
$\left|\frac{b\_{n-1}}{k}+\cdots +\frac{b\_0}{k^n}\right|
\le \frac{nM}{k}
$.

For any $\epsilon > 0$,
if we choose
$k > \frac{nM}{\epsilon}
$,
then
$1-\epsilon
 < 1 + \frac{b\_{n-1}}{k} +$...$+\frac{b\_0}{k^n} 
<1+\epsilon
$,
so we can make the sum
as close to $1$
as we want.
You should learn a bit of logic so that you can properly understand what a mathematical theorem says, especially those with alternating quantifiers such as in analysis. The theorem in this question is essentially asking you to play the following game.

> 
> I make the first move and give you any real numbers $b\_0,...,b\_{n-1}$. You then make the second move and give me a real number $k$. If $\frac{1}{2} < 1 + \frac{b\_{n-1}}{k} + \cdots +\frac{b\_0}{k^n} < \frac{3}{2}$, then you win. If not, I win.
> 
> 
> 

The theorem is true if and only if you have a strategy that ensures that you always win. When you have understood this, then you can try to figure out how to find the $k$, and also it is best for you to express the solution in terms of the game, to make it clear:

> 
> Take any real numbers $b\_0,...,b\_{n-1}$.
> 
> 
> Let $k = \cdots$.
> 
> 
> Then ...
> 
> 
> Thus $\frac{1}{2} < 1 + \frac{b\_{n-1}}{k} + \cdots +\frac{b\_0}{k^n} < \frac{3}{2}$.
> 
> 
> 

Andre has already given you the answer. Mentally check that it really provides a winning strategy. Also, it's not enough to just see that the solution works; you must also figure out how he finds the solution. If you use the game viewpoint, it's quite easy to figure out after you actually play the game with an actual opponent or yourself, a few times.