Why Factorials Make a Difference
A factorial changes rapidly with its index. Recall that \(n!=1\cdot2\cdot3\cdots n\) for a positive integer \(n\), and \(0!=1\). In a power series, factorials in the denominator can make the coefficients small quickly enough that the series converges for every value of its variable.
The useful calculation is often not the coefficient itself, but the ratio of two consecutive coefficients. Many factors then cancel, leaving a simple expression whose limit determines the radius.
This statement assumes the consecutive coefficients used in the ratio are nonzero. The examples below have nonzero coefficients. The formula is a convenient form of the ratio-test result for power series; it tells us about convergence away from the center, while convergence at the center is automatic.
Cancel Consecutive Factorials
The essential identities are \((n+1)!=(n+1)n!\) and, more generally, \((n+k+1)!=(n+k+1)(n+k)!\) for a fixed nonnegative integer \(k\). Thus a quotient of consecutive factorials reduces to one factor, rather than requiring an estimate of the full factorial.
As \(n\) increases, this ratio approaches zero. That shrinking ratio is the reason a coefficient such as \(1/n!\) leads to an infinite radius. The same cancellation works when the factorial is squared or its index is shifted.
Worked Example: A Single Factorial in the Denominator
Worked Example: A Single Factorial in the Denominator
Find the radius of convergence of
The center is \(c=2\), and the coefficient of \((x-2)^n\) is \(a_n=1/n!\). Calculate the ratio of consecutive coefficients:
Taking the limit gives
Therefore the radius is \(R=1/0=\infty\). To check this conclusion directly for any fixed real \(x\), let \(u_n=(x-2)^n/n!\). If \(x\ne2\), then
This limit is less than \(1\), so the ratio test gives absolute convergence. At \(x=2\), every term after the constant term is zero, so the series converges there as well. Thus the series converges for every real \(x\), confirming the infinite radius.
Why the Ratio Calculation Gives the Radius
The coefficient ratio identifies how the terms behave at any fixed distance from the center. Write \(u_n=a_n(x-c)^n\). When \(a_n\) and \(a_{n+1}\) are nonzero,
If the coefficient ratio approaches \(L\), the term ratio approaches \(|x-c|L\). The ratio test gives absolute convergence when this limit is less than \(1\), and divergence when it is greater than \(1\). This establishes the radius formula in the ordinary finite positive case.
For a fixed \(x\), compare \(u_{n+1}\) with \(u_n\), keeping \(|x-c|\) visible.
Compute \(L=\lim |a_{n+1}/a_n|\), usually by cancelling adjacent factorials.
The ratio test compares \(|x-c|L\) with \(1\). If \(L=0\), this is less than \(1\) for every fixed real \(x\), so the radius is infinite.
For \(L=0\), the conclusion follows without leaving any finite distance undecided: for every fixed \(x\), \(|x-c|L=0<1\). The ratio test therefore proves absolute convergence at every such \(x\). This is the complete reason that a coefficient ratio tending to zero gives an infinite radius.
Worked Example: A Squared Factorial
Worked Example: A Squared Factorial
Find the radius of convergence of
The center is \(c=-1\), and \(a_n=1/(n!)^2\). The next coefficient is \(a_{n+1}=1/((n+1)!)^2\). Hence
As \(n\to\infty\), this ratio tends to \(0\), so the radius is \(R=\infty\). For a direct check at any fixed \(x\), the absolute term ratio is
The ratio test therefore gives absolute convergence for every real \(x\). At the center \(x=-1\), the series also converges, with all terms after \(n=0\) equal to zero. The direct test and the coefficient formula agree: the radius is infinite.
Worked Example: A Shifted Factorial
Worked Example: A Shifted Factorial
Find the radius of convergence of
Here \(c=4\) and \(a_n=3^n/(n+2)!\). Advancing the index by one changes \(3^n\) to \(3^{n+1}\) and \((n+2)!\) to \((n+3)!\). Therefore
The limit is
Check at an arbitrary fixed \(x\): the absolute term ratio is
The ratio test proves absolute convergence for every real \(x\), including \(x=4\). Thus the shifted factorial has not changed the conclusion; after cancellation, only the growing factor \(n+3\) remains in the denominator of the ratio.
Factorials in the Numerator: A Necessary Contrast
A factorial does not automatically produce an infinite radius. Its location matters. If a factorial is in the numerator, consecutive coefficients can grow instead of shrink.
Worked Example: A Factorial in the Numerator
Find the radius of convergence of
The coefficients are \(a_n=n!\), so
The radius formula gives \(R=1/\infty=0\). To check the divergence away from the center, for any fixed \(x\ne0\) the term ratio is
Eventually this ratio is greater than \(1\), so the terms do not tend to zero and the series diverges. At \(x=0\), only the constant term is nonzero, so the series converges at its center. The radius is still \(0\), as confirmed by divergence at every nonzero distance from the center.
Common Mistake: Comparing Factorials Without Cancelling
A common error is to see a factorial and immediately declare that the radius is infinite. For example, with coefficients \(a_n=n!\), it is wrong to write \(a_{n+1}/a_n\to0\) merely because the expression contains factorials. The correct calculation is
not zero. The corrected conclusion is that the radius is zero, not infinite. Always form the quotient in the order \(a_{n+1}/a_n\), cancel exactly, and only then take the limit. For a reciprocal factorial, the quotient reverses and tends to zero; for a factorial in the numerator, it grows.
What an Infinite Radius Tells You
A radius of \(R=\infty\) means the power series converges absolutely for every real \(x\). There is no finite boundary point at which a separate endpoint test is needed. This does not mean every series containing a factorial has this property: the coefficient ratio must actually tend to zero. Squared factorials and fixed shifts can be handled by the same consecutive-term calculation, but the conclusion depends on the resulting limit.
Check Your Understanding
For each question, show the consecutive-coefficient ratio before stating the radius.
- What is the ratio \(\left|a_{n+1}/a_n\right|\) when \(a_n=1/(n!)^3\), and what does its limit imply?
- For \(a_n=2^n/(n+1)!\), simplify \(\left|a_{n+1}/a_n\right|\) and find the radius.
- Why does the series \(\sum_{n=0}^{\infty}(x-c)^n/(n+4)!\) have an infinite radius?
- For coefficients \(a_n=(n+2)!\), does the coefficient ratio tend to zero or infinity? What is the radius?
- What does an infinite radius say about convergence at finite real values of \(x\)?
