Is there a way to evaluate this infinite series? $\sum_{n=1}^{\infty} \frac {1}{3n^2+2}$
In my calculus 2 class, we were testing the convergence of different series by using the direct comparison test and confirmed that this one converges, but I tried to evaluate it afterward to no avail. I factored the denominator then used partial fraction decomposition to separate the roots, but I'm not sure that got me any closer. Any help would be appreciated. 
 A: There is a known rational function power series (see Wikipedia) which we can manipulate your summation into:
$$\sum_{n=0}^\infty \frac{1}{n^2 + a^2} = \frac{1 + a \pi \coth(a \pi)}{2a^2}$$
From your power series, we will add and subtract an $n=0$ term first, pulling the added one into the summation:
$$S = \sum_{n=1}^\infty \frac{1}{3n^2 + 2} = - \frac 1 2 + \sum_{n=0}^\infty \frac{1}{3n^2 + 2}$$
Next, we pull out a factor of $1/3$ from the summation:
$$S = - \frac 1 2 + \frac 1 3 \sum_{n=0}^\infty \frac{1}{n^2 + 2/3}$$
Apply the initial formula we referenced with $a=\sqrt{2/3}$; with some simplifying,
$$S =  \frac{-1 + \pi \sqrt{2/3} \cdot \coth( \pi \sqrt{2/3})}{4} \approx 0.398906832$$
Of course, justifying the original power series is probably well beyond Calculus II levels, as noted in the comments of your question, but I'm not sure how else this series might be evaluated. Verifying the convergence isn't troublesome as I imagine you noticed - finding the actual value, however, can be quite the task for many infinite sums of rational functions.
Of course, there might be other methods, who knows. I wouldn't get my hopes up on the matter though.
A: The series can be estimated directly by a partial sum
$$ \sum_{n=1}^{a-1} \frac {1}{3n^2+2}$$
and estimating the error by integral test
$$\int_a^\infty \frac {1}{3x^2+2}dx\le \sum_{n=a}^{\infty} \frac {1}{3n^2+2}\le f(a)+\int_a^\infty \frac {1}{3x^2+2}dx$$
For example for $a=10$ we obtain
$$ \sum_{n=1}^{9} \frac {1}{3n^2+2}\approx 0.364$$
$$0.0333 \approx\int_{10}^\infty \frac {1}{3x^2+2}dx\le \sum_{n=10}^{\infty} \frac {1}{3n^2+2}\le f(a)+\int_{10}^\infty \frac {1}{3x^2+2}dx\approx 0.0363$$
therefore
$$ \sum_{n=1}^{\infty} \frac {1}{3n^2+2}\in [0.3973,0.4003]$$
