Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series
Key Questions

In most cases, an alternation series
#sum_{n=0}^infty(1)^nb_n# fails Alternating Series Test by violating#lim_{n to infty}b_n=0# . If that is the case, you may conclude that the series diverges by Divergence (Nth Term) Test.
I hope that this was helpful.

Alternating Series Test
An alternating series
#sum_{n=1}^infty(1)^n b_n# ,#b_n ge 0# converges if both of the following conditions hold.#{(b_n ge b_{n+1} " for all " n ge N),(lim_{n to infty}b_n=0):}#
Let us look at the posted alternating series.
In this series,
#b_n=1/sqrt{3n+1}# .#b_n=1/sqrt{3n+1} ge 1/sqrt{3(n+1)+1}=b_{n+1}# for all#n ge 1# .and
#lim_{n to infty}b_n=lim_{n to infty}1/sqrt{3n+1}=1/infty=0# Hence, we conclude that the series converges by Alternating Series Test.
I hope that this was helpful.

Alternating Series Test states that an alternating series of the form
#sum_{n=1}^infty (1)^nb_n# , where#b_n ge0# ,
converges if the following two conditions are satisfied:
1.#b_n ge b_{n+1}# for all#n ge N# , where#N# is some natural number.
2.#lim_{n to infty}b_n=0# Let us look at the alternating harmonic series
#sum_{n=1}^infty (1)^{n1}1/n# .
In this series,#b_n=1/n# . Let us check the two conditions.
1.#1/n ge 1/{n+1}# for all#n ge 1#
2.#lim_{n to infty}1/n=0# Hence, we conclude that the alternating harmonic series converges.
Questions
Tests of Convergence / Divergence

Geometric Series

Nth Term Test for Divergence of an Infinite Series

Direct Comparison Test for Convergence of an Infinite Series

Ratio Test for Convergence of an Infinite Series

Integral Test for Convergence of an Infinite Series

Limit Comparison Test for Convergence of an Infinite Series

Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series

Infinite Sequences

Root Test for for Convergence of an Infinite Series

Infinite Series

Strategies to Test an Infinite Series for Convergence

Harmonic Series

Indeterminate Forms and de L'hospital's Rule

Partial Sums of Infinite Series