Why Exponential and Polynomial Factors Appear Together
Series terms often combine a power of the index with a fixed number raised to that index. The power \(n^p\) changes with \(n\), but so does the exponential factor \(c^n\). It is not enough to look at the size of one factor in isolation: the behavior of the whole term matters.
The ratio test is especially useful for these combinations. When we compare successive absolute values, the exponential contributes a constant factor, while the ratio of successive polynomial factors approaches one. The resulting limit makes the role of the exponential factor clear.
The Ratio Test for These Terms
For a series \(\sum a_n\), the ratio test examines the limit of \(\lvert a_{n+1}\rvert/\lvert a_n\rvert\), when that limit exists. If the limit is less than one, the series converges absolutely. If it is greater than one, the series diverges. A limit equal to one does not decide convergence or divergence.
exists, then \(\sum a_n\) converges absolutely if \(L<1\), and diverges if \(L>1\). If \(L=1\), the ratio test gives no conclusion.
The theorem applies to the absolute values, so the sign of \(c\) does not change the outcome. In the calculations below, \(|c|\) therefore appears in the successive-term ratios.
Deriving the Ratios
First consider \(a_n=n^p c^n\). Since \(|a_n|=n^p|c|^n\), its successive absolute-value ratio is
For the other form, let \(b_n=n^p/c^n\). Then \(|b_n|=n^p/|c|^n\), and
For every fixed real \(p\), the factor \(\left(1+\frac{1}{n}\right)^p\) tends to one. Indeed, \(1+\frac{1}{n}\) tends to one, and the function \(x\mapsto x^p\) is continuous for \(x>0\). Therefore, the ratio-test limits are
This derivation explains why the polynomial factor does not change the limiting ratio: its contribution is the ratio of consecutive powers, which approaches one. The fixed exponential factor remains in the limit.
Classification at a Glance
| Term form | Ratio-test limit | Conclusion from the ratio test |
|---|---|---|
| \(n^p c^n\) | \(|c|\) | Absolute convergence if \(|c|<1\); divergence if \(|c|>1\) |
| \(n^p/c^n\) | \(1/|c|\) | Absolute convergence if \(|c|>1\); divergence if \(|c|<1\) |
| Either form with \(|c|=1\) | \(1\) | The ratio test is inconclusive |
The direction reverses between the two forms. In \(n^p c^n\), a base with absolute value less than one makes the exponential factor decay. In \(n^p/c^n\), that decay occurs when \(|c|>1\), because division by \(c^n\) is equivalent in absolute value to multiplication by \((1/|c|)^n\).
Worked Examples
Worked Example: A Polynomial Factor with Exponential Decay
Determine whether \(\sum_{n=1}^{\infty} n^3(2/5)^n\) converges.
Set \(a_n=n^3(2/5)^n\). These terms are positive, so their absolute values are the terms themselves. The next term is \(a_{n+1}=(n+1)^3(2/5)^{n+1}\). Thus,
As \(n\) tends to infinity, \(\left(1+\frac{1}{n}\right)^3\) tends to one. Therefore,
The ratio test shows that the series converges absolutely. This agrees with the general condition \(|c|<1\) for \(n^p c^n\). The computed limit, \(2/5\), is indeed less than one, so the convergence conclusion follows.
Worked Example: A Polynomial Divided by an Exponential
Determine whether \(\sum_{n=1}^{\infty} n^2/(3/2)^n\) converges.
Let \(b_n=n^2/(3/2)^n\). The next term is \(b_{n+1}=(n+1)^2/(3/2)^{n+1}\). Dividing successive terms gives
The polynomial ratio tends to one, so
The series converges absolutely by the ratio test. As a check on the direction of the factor, division by \((3/2)^n\) contributes \(1/(3/2)=2/3\), not \(3/2\), to the limiting ratio. Since \(2/3<1\), the test gives convergence.
Worked Example: Exponential Growth Overcomes Polynomial Decay
Determine whether \(\sum_{n=1}^{\infty} n^{-4}(3/2)^n\) converges.
Here \(a_n=n^{-4}(3/2)^n\). Its successive absolute-value ratio is
The factor \(\left(1+\frac{1}{n}\right)^{-4}\) tends to one. Hence,
The ratio test shows that the series diverges. In particular, a negative power of \(n\) does not reverse the conclusion when the fixed exponential factor has base \(3/2>1\). The limit is greater than one, as required for the divergence conclusion.
The Boundary Case and a Common Mistake
When \(|c|=1\), both ratio formulas give a limit of one, regardless of the fixed real value of \(p\). That is the ratio test’s inconclusive case. It does not mean that the series converges, and it does not mean that the series diverges. Another argument is needed to decide a particular boundary series.
Worked Example: A Ratio Limit of One Is Not a Conclusion
Consider \(\sum_{n=1}^{\infty} n^2\), which has the form \(n^p c^n\) with \(p=2\) and \(c=1\). Its successive-term ratio is
The ratio test gives no conclusion because its limit equals one. However, the terms \(n^2\) do not tend to zero, so the series diverges by the necessary condition that the terms of a convergent series must tend to zero. This separate check establishes divergence; the ratio-test limit alone did not.
A frequent error is to invert the exponential factor when calculating a ratio. For example, for \(b_n=n^3/2^n\), it is incorrect to write
The factor from the denominator is \(2^n/2^{n+1}=1/2\), not \(2\). The correct calculation is
Thus the ratio test proves absolute convergence. Checking whether the exponential is in the numerator or denominator before simplifying helps prevent this reversal.
Using the Result Efficiently
Determine whether the term is \(n^p c^n\) or \(n^p/c^n\), and note that \(c\ne0\).
Cancel the powers carefully; a denominator \(c^n\) contributes \(1/|c|\), not \(|c|\).
The polynomial ratio \(\left(1+\frac{1}{n}\right)^p\) tends to one, leaving \(|c|\) or \(1/|c|\).
A limit below one proves absolute convergence, a limit above one proves divergence, and a limit equal to one requires a different test.
These calculations capture the central distinction: for a fixed base, the exponential factor controls the ratio-test limit, while the polynomial factor contributes a multiplier approaching one. The ratio test therefore gives a direct decision away from the boundary \(|c|=1\), even when \(p\) is positive, zero, or negative.
Check Your Understanding
For each question, show the successive-term ratio and state precisely what the ratio test concludes.
- What limit does the ratio test produce for terms of the form \(n^p c^n\)?
- For terms of the form \(n^p/c^n\), why does the limiting ratio involve \(1/|c|\)?
- What does the ratio test conclude when the limiting ratio is greater than one?
- Why does the polynomial factor not alter the limiting ratio for fixed \(p\)?
- What can the ratio test conclude when \(|c|=1\), and what does that outcome mean?
