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 [sequence]s (especially the ones filled with nasty [exponent]s):

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].

ratio test (idea)