Look up Mathematica's Help and figure out how to enter infinite series and test their convergence.
Then use what you learned to check two "easy" series (e.g., p-series), one of which you already know converges, and the other you already know diverges. See this page Hints for Using Mathematica if you need a Mathematica refresher.
Now do Sec 9.4: 42.
Hint for (a): Mathematica gives a complicated looking answer for the series:
-(1/3) RootSum[2 + 3 #1 + 3 #1^2 + #1^3 &, PolyGamma[0, -#1]/(1 + 2 #1 + #1^2) &]
That's just because it's trying to give an exact answer, rather than a decimal approximation. But it doesn't say that the series diverges --- so we can conclude it converges.
Now do parts (b)-(e). For part (c), you can either use a spreadsheet to create a table, or look up the Mathematica function Table[ ] (feel free to ask me for help).
After you're done with (e), tell Mathematica to give you a numerical approximation for the value of the infinite sum, by using the function N[ ].
What did you get? The "i" stands for Sqrt[-1]; note that it's multiplied by 0. Ask me what this means if you're not familiar with or have forgotten about complex numbers.