How to test convergence of related series from \(\sum c_n 4^n\)

Expert intro: This kind of question is really about what the known convergence at one point does and does not tell you. A single convergent value of a power series can support some substitutions, but not all of them.

Detailed walkthrough

We are told that

$\sum_{n=0}^{\infty} c_n 4^n$

is convergent. We must decide whether this implies convergence of:

(a) $\sum_{n=0}^{\infty} c_n(-2)^n$

Rewrite the terms as

$c_n(-2)^n=c_n4^n\left(-\frac12\right)^n.$

So the new series is the old coefficient series evaluated at a different input. But knowing convergence at $x=4$ alone does not guarantee convergence at $x=-2$.

Why not? A power series can converge at one point outside a smaller radius only under special circumstances, and the sign change matters. Without knowing the radius of convergence or more data about $c_n$, there is no valid conclusion.

So the answer is: cannot conclude.

(b) $\sum_{n=0}^{\infty} c_n(-4)^n$

Here,

$c_n(-4)^n=c_n4^n(-1)^n.$

This is an alternating version of the known convergent series. But again, convergence of

$\sum c_n4^n$

does not automatically imply convergence of

$\sum c_n(-4)^n.$

A sign change may completely alter convergence behavior.

So the answer is also: cannot conclude.

Final answers

• (a) No, we cannot conclude it is convergent.
• (b) No, we cannot conclude it is convergent.

💡 Pitfall guide

The trap is assuming that if a power series converges at one point, then it must converge at any point with the same absolute value. That is false. The radius of convergence is centered at the series variable, and you need more than one data point to compare locations reliably.

It is also easy to confuse pointwise convergence of a specific series with absolute convergence of the coefficients themselves. Those are not the same thing.

🔄 Real-world variant

If you were additionally told that $\sum c_n x^n$ has radius of convergence $R>4$, then both $x=4$ and $x=-4$ would lie inside the disk of convergence, so both series would converge. If $R=4$, then the endpoints would need separate checking, and neither conclusion would follow just from the given information.

🔍 Related terms

radius of convergence, power series, conditional convergence