Question: There are a variety of tests used to determine the convergence of infinite series.

Task:

1. Determine the convergence or divergence of the geometric series \[\sum\limits_{j=1}^{\infty }{=4(-\frac{1}{3}}{{)}^{j-1}}\] showing all work.
2. Use the n th term test to determine whether the series \[\sum\limits_{j=1}^{\infty }{\frac{{{j}^{2}}+1}{{{j}^{2}}}}\] converges or diverges, showing all work.
3. Use the integral test to determine whether the \[\sum\limits_{k=1}^{\infty }{\frac{1}{3k+1}}\] converges or diverges, showing all work.
4. Determine the convergence or divergence of the p -series \[\sum\limits_{n=1}^{\infty }{\frac{1}{{{n}^{(In3)}}}}\] showing all work.

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