Determine if the given series converges and if so find its convergence $$\lim_{N \to \infty}\sum_{n=0}^{N} \frac{3^{1-2n}}{n^2+1}$$
$$\lim_{n \to \infty}\frac{a_{n+1}}{{a_n}} = \lim_{n \to \infty}\frac{3^{1-2(n+1)}}{(n+1)^2+1}\frac{n^2+1}{3^{1-2n}}$$
this comes out to be:
$$\lim_{n \to \infty} \frac{n^2+1}{9[(n+1)^2+1]}=\frac{1}{9}$$
Is my work correct? I just started Calculus II. Could this have been solved using another method?
 A: Before going for complicated calculations, always think about simplifying with rough inequalities...
Note that this is a series with positive terms so we can apply comparison theorem (i.e. partial sums are $\nearrow$ and bounded by another series that converges, thus converge too).
Indeed:


*

*$3^{1-2n}\le 3$

*$\dfrac 1{n^2+1}\le \dfrac 1{n^2}$
And the series $\sum\frac 3{n^2}$ is convergent (you recognize a Riemann series with exponent $>1$, usual way to prove cv is to bound it by the corresponding integral).
A: This is correct, though you will want to state that you are using the ratio test, and since the limit is less than 1 in magnitude, it converges absolutely.
As for alternatives, the direct comparison test is very powerful, and actually the test from which most of the other tests implicitly use, if you can use it. The goal is recognizing similar series that you already know how to solve. Note simple inequalities such as
\begin{align}0\le\frac1{n^2+1}&\le\color{green}{\frac1{n^2}}\le\color{#3377ff}1\\0\le3^{1-2n}&\le\color{red}3\end{align}
From which we may derive inequalities such as
$$0\le\sum_{n=0}^\infty\frac{3^{1-2n}}{n^2+1}\le\frac{3^{1-2(0)}}{0^2+1}+\sum_{n=1}^\infty\frac{\color{red}3}{\color{green}{n^2}}$$
as was found in zwim's answer. From here we can argue the convergence using tests such as the p-series test, integral test, Cauchy's condensation test, etc.
We could also argue that we have
$$0\le\sum_{n=0}^\infty\frac{3^{1-2n}}{n^2+1}\le\sum_{n=0}^\infty\color{#3377ff}1\cdot3^{1-2n}=\frac3{1-3^{-2}}$$
which gives us a geometric series with first term $3$ and common ratio $3^{-2}$.
This will require more experience, as you have to be able to draw the connections between your series and other series, but once you have that it is very useful. Consider looking at all of the convergence test problems you have been given and try to see if any look familiar. Then try to make inequalities between them.
