Two Extreme Behaviors
A power series can converge for every real input, or it can fail to converge at every input except its center. These behaviors may seem surprising: both are determined by how quickly the coefficients change from one term to the next.
The ratio test makes this comparison direct. For a power series centered at \(c\), the factor \((x-c)^n\) contributes a fixed multiplier when \(x\) is fixed, while the coefficient ratio describes how the coefficients themselves grow or shrink. If the coefficients shrink sufficiently quickly, the series can converge no matter how far \(x\) is from \(c\). If they grow sufficiently quickly, even a small nonzero distance from \(c\) can make the terms fail to approach zero.
Reading the Ratio Test for a Power Series
Fix a real input \(x\) and let \(u_n=a_n(x-c)^n\) be the \(n\)th term. When the terms are eventually nonzero, their successive absolute-value ratio is
The ratio test says that if this ratio tends to a limit \(L<1\), the series converges absolutely; if it tends to \(L>1\), the series diverges. A limit of \(1\) gives no conclusion. In the power-series setting, the input contributes the fixed factor \(|x-c|\), so the behavior of the coefficient ratios is especially informative.
Why the Coefficient Ratio Determines These Cases
The result follows by applying the ratio test at a fixed input rather than by estimating the full series in some other way. The factor \(|x-c|\) stays constant as \(n\) increases. Thus, if the coefficient ratio tends to zero, the ratio of successive terms tends to zero at every fixed input. If the coefficient ratio tends to infinity, that term ratio tends to infinity at every fixed input other than the center.
For a fixed \(x\ne c\), its absolute value is \(\left|\frac{a_{n+1}}{a_n}\right||x-c|\), whenever the coefficient ratio is defined.
If the coefficient ratio tends to \(\rho\), the term ratio tends to \(\rho|x-c|\). This uses that \(|x-c|\) is fixed.
A limit below \(1\) gives absolute convergence; a limit above \(1\) gives divergence. When \(\rho=0\), the first condition holds at every real input.
At \(x=c\), every term with \(n\ge1\) is zero, so the series reduces to its constant term \(a_0\) and converges.
This also justifies the finite-positive case: if \(0<\rho<\infty\), then \(\rho|x-c|<1\) exactly when \(|x-c|<1/\rho\), and \(\rho|x-c|>1\) exactly when \(|x-c|>1/\rho\). At equality, the ratio limit is \(1\), so the ratio test alone cannot decide convergence.
Factorials in the Denominator: Infinite Radius
A factorial in a denominator often makes successive coefficients shrink toward zero. Since \((n+1)!=(n+1)n!\), the coefficient ratio for \(1/n!\) is \(1/(n+1)\), which tends to zero. The fixed factor \(|x-c|\) cannot prevent the resulting term ratio from tending to zero for any particular real input.
Worked Example: A Factorial Denominator
Determine where the power series \(\sum_{n=0}^{\infty}\frac{(x-2)^n}{n!}\) converges.
For a fixed \(x\), set \(u_n=\frac{(x-2)^n}{n!}\). If \(x\ne2\), the terms are nonzero, and their successive absolute-value ratio is
For every fixed real \(x\), the numerator \(|x-2|\) is constant and \(n+1\) grows without bound. Therefore
The ratio test gives absolute convergence for every \(x\ne2\). At \(x=2\), the series is \(1+0+0+\cdots=1\), so it converges there as well. Thus the series converges for every real input and has infinite radius. Checking the ratio confirms the conclusion: for instance, at \(x=5\), the ratio is \(3/(n+1)\), which tends to \(0\).
Factorials in the Numerator: Zero Radius
A factorial in a numerator has the opposite effect. Because \((n+1)!/n!=n+1\), the coefficient ratio grows without bound. At any fixed nonzero distance from the center, multiplying that ratio by the distance still produces a limit of infinity. The ratio test then gives divergence.
Worked Example: A Factorial Coefficient
Determine where \(\sum_{n=0}^{\infty} n!(x+3)^n\) converges.
For \(x\ne-3\), let \(u_n=n!(x+3)^n\). The successive absolute-value ratio is
Since \(|x+3|\) is a fixed positive number when \(x\ne-3\),
The ratio test therefore gives divergence for every \(x\ne-3\). At the center \(x=-3\), all terms after the constant term vanish, and the series equals \(0!=1\). It converges only at its center, so its radius is zero. As a numerical check, at \(x=-2\) the term ratio is \(n+1\); for \(n=2\) it is \(3\), and it continues to increase, confirming that it does not tend to a value below \(1\).
Even Faster-Growing Coefficients
Factorials are not the only coefficients that force a zero radius. A coefficient such as \(2^{n^2}\) has successive ratio \(2^{(n+1)^2-n^2}=2^{2n+1}\), which itself grows rapidly. The same test applies: calculate the quotient of consecutive coefficients carefully, then include the fixed distance from the center.
Worked Example: A Rapidly Growing Coefficient
Determine where \(\sum_{n=0}^{\infty}2^{n^2}x^n\) converges.
For \(x\ne0\), set \(u_n=2^{n^2}x^n\). Then
Here \((n+1)^2-n^2=2n+1\), so for each fixed \(|x|>0\), the ratio tends to infinity. The ratio test gives divergence at every \(x\ne0\). At \(x=0\), the series reduces to its constant term \(2^0=1\), so it converges there. Its radius is zero. For a check at \(x=\frac14\), the ratio is \(2^{2n+1}/4=2^{2n-1}\); already at \(n=1\) this ratio is \(2\), and it increases without bound, in agreement with the divergence conclusion.
A Common Ratio-Reversal Mistake
When a factorial appears in the denominator, it is easy to reverse the coefficient quotient and mistakenly conclude that the coefficients grow. For the series in the first example, the coefficient is \(a_n=1/n!\), so the ratio must be the next coefficient divided by the current one:
A wrong calculation would be \(\frac{(n+1)!}{n!}=n+1\), which reverses the quotient and describes the ratio for coefficients \(n!\), not \(1/n!\). The correction matters: for \(1/n!\), the coefficient ratio tends to zero and the series has infinite radius; for \(n!\), it tends to infinity and the radius is zero. Before interpreting a factorial pattern, verify which coefficient is in the numerator of \(a_{n+1}/a_n\).
What These Extreme Cases Tell You
The ratio test turns coefficient growth into a statement about the whole real line. A limiting coefficient ratio of zero overwhelms every fixed distance from the center, producing convergence everywhere. An infinite limiting coefficient ratio overwhelms every nonzero distance, leaving convergence only at the center. In both cases, the center is confirmed directly because the series reduces to one term.
For a finite positive coefficient-ratio limit, the same calculation identifies a finite positive radius, but it does not decide convergence at points where the ratio limit for the terms is exactly \(1\). Keep that limitation in view: a ratio test result of \(1\) is inconclusive, not proof of either convergence or divergence.
Check Your Understanding
For each question, use successive ratios and state any input where the ratio test does not apply directly.
- For \(\sum_{n=0}^{\infty}\frac{(x-1)^n}{(n+2)!}\), what is the limit of the absolute-value term ratio at a fixed \(x\ne1\), and what does it imply?
- For \(\sum_{n=0}^{\infty}(2n)!(x+1)^n\), calculate the successive absolute-value term ratio and determine its behavior when \(x\ne-1\).
- For coefficients \(a_n=5^{-n}\), find \(\lim_{n\to\infty}|a_{n+1}/a_n|\). What radius does the ratio test predict for a power series with these coefficients?
- For \(\sum_{n=0}^{\infty}4^{n^2}(x-2)^n\), calculate the coefficient ratio and determine where the series converges.
- Why does a coefficient-ratio limit of \(0\) imply convergence at every fixed real input, while a limit of infinity only gives divergence away from the center?
