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Mathematical Expression Editor

Some infinite series can be compared to geometric series.

As mathematicians, we are explorers. We explore the implications of seemingly
simple quantitative facts.

Consider the infinite series .

Let be the formula for the terms in sequence whose sum we are trying to find. We
can observe something interesting when is large by looking at the ratio
.

Notice that by treating the division by as multiplication by its reciprocal,

After rearranging a little, we have

and we can now finish computing the limit.

Thus, when is large, is pretty close to halfdouble of . So, when we choose a very large whole number , should be approximately , and
this should occur for all successive terms in the sequence . That is,

for all .

We can then observe the following.

We thus have that

In words, the tail of the sequence is “approximately” a geometric series with ratio
.

Does a geometric series with ratio converge or diverge?

convergediverge

Given your answer above, do you suspect that the original sum converges or
diverges?

convergediverge

As it turns out, the above ideas can be formalized and written as a theorem. The
proof of this theorem is slightly beyond the scope of the course, but really only
involves introducing formal mathematical language to make the above observations
precise.

The Ratio Test Consider and set

If , then the series converges.

If or is infinite, then the series diverges.

If , the test is inconclusive; the series could diverge or converge.

Note that this is easy to remember if you just use the following heuristic.

If the ratio test gives a limit of , then the series is like a geometric
series of ratio .

The case of is an “edge” case, and can go either way, which we will demonstrate with
specific examples in a bit.

Now that you have the basic idea, we give examples showing the following.

Convergence by the ratio test.

Divergence by the ratio test.

A divergent series for which the ratio test is inconclusive.

A convergent series for which the ratio test is inconclusive.

In these examples, pay attention to how the ratio of different types of terms behave
and simplify.

Consider . Discuss the convergence of this series.

We
will attempt to use the ratio test. Setting , we compute

So, the ratio test guarantees that is convergent guarantees that is divergentgives no information in this case, but we know the series is convergent through some
other methodgives no information in this case, but we know the series is divergent
through some other method.

So the series is convergent by the ratio test. Note that this shows that grows much
faster than the exponential function .

Consider . Discuss the convergence of this series.

We
will attempt to use the ratio test. Setting , we compute

So, the ratio test guarantees that is convergent guarantees that is divergentgives no information in this case, but we know the series is convergent through some
other methodgives no information in this case, but we know the series is divergent
through some other method.

We now turn to two examples where the ratio test will be inconclusive.

Consider . Discuss the convergence of this series.

Note that this question is slightly
ridiculous; since , the series will diverge by the divergence test. However, if we try to
use the ratio test, what happens?

So, the ratio test guarantees is convergentguarantees is divergentis
inconclusive.

Consider . Discuss the convergence of this series.

We will attempt to use the ratio test. Setting , notice

We could do some algebra here, but notice that the dominant term in both the
numerator and denominator will be . As such, we can conclude that the limit is
without further computation.

The ratio test guarantees is convergentguarantees is divergentis
inconclusive.

In order to determine what happens here, we can actually use partial fraction
decomposition to show that

As such, we can write and recognize this as a telescoping series. By setting and
writing out several terms (as done in an earlier section), we can find an explicit
formula

Since , we conclude that the series converges to 1.

It is important that examples illustrating the final two behaviors exist, because it
shows that the ratio test really is inconclusive in the case .

In the previous examples, we studied sequences whose terms involved products of
factorials, exponentials, and polynomials. One interesting observation to note is that
the ratio of polynomials in the examples above did not affect the value of
and this was perhaps most readily evident in the wildly divergent series
.

As it turns out, this fact will be important in an observation we make later, so we
dignify it with a theorem here.

Suppose that is a polynomial. Then,

This theorem guarantees that if is a sequence whose terms involve products or
quotients of polynomials, these polynomial terms will have no effect on the limit that
is computed using the ratio test.

Let and . According to the theorem, and should be equal. To verify this, notice
that

and also that

This example also motivates an important fact. Notice that for each term with , we
have , but the limits necessary to compute for the ratio test are equal. As such, we
observe the following.

The ratio test can only tell us that a series converges; it cannot give the value to
which the series converges.

The theorem also leads to a very important observation about when we
should attempt to use the ratio test. A series must have a term that grows
at least exponentially in order for the Ratio Test to have a chance to be
conclusive!

More explicitly, let , , and consider the growth rates results for sequences:

Without performing any calculations, determine which of the following series would
the Ratio Test would be conclusive. That is, which of the following would
either converge or diverge as a consequence of the Ratio Test?

Summary

We summarize the important results about the ratio test. Consider a series
.

If is a polynomial, , so the ratio test will only be conclusive if has a factor
that grows at least exponentially (according to the growth rates results).

To use the test, set

If , then the series converges.

If or is infinite, then the series diverges.

If , the test is inconclusive; the series could diverge or converge.

In the case where a series converges, the ratio test gives no information about
the value of the series.