The exact value of $\sum_{n=1}^{\infty}\frac{n^2+n+1}{3^n}$ What is the value of :
$$\sum_{n=1}^{\infty}\frac{n^2+n+1}{3^n}$$

In most practical applications, you can just ask Mathematica, and it will tell you it's $\frac{11}{4}$.
If you want to arrive at the formula in a more rigorous way, you can do the following: 
Consider the function $f$: $$f(x) = \frac{1}{1 - x} = \sum_{n=0}^\infty x^n$$
An initial observation is that $f(\frac{1}{3}) = \sum_{n=0}^\infty \frac{1}{3^n}$ so it looks a little like your sum. Now, consider $f'$:
$$f'(x) = \frac{1}{(1 - x)^2} = \sum_{n=1}^\infty n x^{n-1} = \sum_{n=0}^\infty (n+1) x^{n} $$
When you plug in $\frac{1}{3}$ again, you find $f'(\frac{1}{3}) = \sum_{n=0}^\infty \frac{n+1}{3^n}$. Finally, consider $f''$:
$$f''(x) = \frac{2}{(1 - x)^3} = \sum_{n=0}^\infty (n+2)(n+1) x^{n} = \sum_{n=0}^\infty (n^2 + 3n + 2) x^{n}$$
When you plug in $\frac{1}{3}$ once more, you find $f''(\frac{1}{3}) = \sum_{n=0}^\infty \frac{n^2 + 3n + 2}{3^n}$. Now, all you have to do is to express $n^2 + n +1$ as:
$$ n^2 + n + 1 = (n^2 + 3n + 1) - 2 (n+1) + 1 $$
so you get that the sought sum $S$ is:
$$ S = f''\left(\frac{1}{3}\right) - 2 f'\left(\frac{1}{3}\right) - f\left(\frac{1}{3}\right)$$
The remaining computation is not pleasant, but it is definitely doable, and does not involve any new creative ideas.

I was a little too careless: the computation I did was for a sum raging from $n=0$ rather than $n=1$ as in the problem. It is however easy to mend - just subtract the initial term, which is just $\frac{1}{3}$.
