If the basic convergence tests (see the infinite series node for a list of essential tests and tips) do not yield anything and you cannot easily apply the Cauchy condition to determine the convergence of an infinite series, then you might want to give d'Alembert's ratio test a try.

Although "less powerful", it is much easier to apply, when facing certain complex sequences (especially the ones filled with nasty exponents):

Let there be a series **Σu**_{n} where (**u**_{n}) is a **strictly positive** sequence (**u**_{n > }0 for all **n**).

If you can find a real **k **positive and **strictly less** than 1 ( **k ∈ [0,1[ **) and an integer **N** such as:

** u**_{n+1
}------ < k for all **n ≥ N**

u_{n}

Then **Σu**_{n} **converges**.

On the other hand:

If you can find a real **r** **strictly greater** than 1 ( **r > 1 **) and an integer **N** such as:

** u**_{n+1
}------ > r for all **n ≥ N**

u_{n}

Then **Σu**_{n} **diverges**.

The "intuitive" way of putting this test (please bear with the imprecision inherent to an attempt to explain rather abstract mathematic concepts into layman's terms) is that, if after a certain rank (**N**), the sequence's terms keep getting a bit "closer" together with *each* increment of **n** (hence the importance of the strict relations), the series will converge. While if each term in the sequence is a bit "farther" from (or equal to) the previous one, the sequence will never "stabilize" and will therefore diverge.

The proof to this test relies on the Cauchy condition.

**What's Next?**

If this test does not yield any conclusive result on the convergence of your series, you might want to look at Raabe's test or Cauchy Condition (and make sure you have previously tried all the trivial conditions listed under the infinite series node).

*Extra Bonus Mathematics Trivia: I bet you did not know that Mr. Jean Le Rond d'Alembert also happened to be co-author (along with Denis Diderot) of L'Encyclopedie, an 18th century attempt at collecting an exhausive and detailed catalog of all forms of human knowledge that can arguably be deemed the first serious precursor to E2. Well now you know.*