The **limit comparison test** is an easy way to compare the limit of the terms of one series with the limit of terms of a known series to check for convergence or divergence. It may be one of the most useful tests for convergence.

The limit comparison test (**LCT**) differs from the direct comparison test. In the comparison test, we compare series elements term-by-term. In the LCT we compare the limits on the sizes of the terms as **n → ∞**.

**Proof**: Let **x** and **y** be positive numbers and let's further say that **L** is between **x** and **y: x < L < y**. Now for large **n**, the ratio **a _{n}/b_{n}** is very close to

Now if we multiply that inequality through by **b _{n}**, we get

Now if **Σb _{n}** converges, so does

By the same logic, if **Σb _{n}**

We set up the limit like this. In the second step we multiply by the reciprocal of the denominator (the rules of basic algebra never change!):

Now to get a better look at that limit, divide every term by **3 ^{n}**:

which reduces to

Now this limit is easy to evaluate. As **n → ∞** the fraction equals 1, which is greater than zero, so the series converges by limit comparison with a known convergent series.

**Solution**: The first thing we need to do in such problems is to find some approximation of the series. For large **n** (in which case the 1 in the numerator doesn't matter), this series is approximately equal to the divergent p-series **1/n ^{1/2}**, so we can use that for the limit comparison test, in which we'll guess that the series is divergent.

Here's the limit expression. It reduces nicely to an easy-to-evaluate limit:

By dividing everything by **n**, we get to a limit that's easier to see:

The limit is greater than zero, and we compared our series to a divergent series, therefore the series we tested is divergent.

This series could also have been compared directly by asking whether its terms are, term-by-term, greater than those of the divergent series with terms **1/n ^{1/2}**.

Cross multiplication gives

Subtracting terms from both sides gives

,

which is true for all **n ≥ 1**, as defined in the series.

Determine whether these series converge using the limit comparison test (LCT).

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