Read Theorem 6.2 in the textbook
Links to an external site.
.

Suppose that
∞
∑

n=0cnxn is a power series with interval of convergence (−3,3).

Suppose that
∞
∑

n=0dnxn is a power series with interval of convergence [−2,2] .

What is the interval of convergence of the series
∞
∑

n=0(cnxn−dnxn)?

Question

Read Theorem 6.2 in the textbook
Links to an external site.
.

Suppose that 
∞
∑

n=0cnxn  is a power series with interval of convergence (−3,3).

Suppose that 
∞
∑

n=0dnxn  is a power series with interval of convergence [−2,2] .

What is the interval of convergence of the series 
∞
∑

n=0(cnxn−dnxn)?

BlackTom AI · Accepted Answer

The problem asks for the interval of convergence of the series ∑ (c_n x^n − d_n x^n) = ∑ (c_n − d_n) x^n, given information about two separate power series.

First, recall the basic facts: a power series ∑ a_n x^n has an interval (or radius) of convergence determined by its radius R, and the endpoints may or may not be included depending on the coefficients. Here, the series ∑ c_n x^n has interval of convergence (−3, 3), so its radius is R1 = 3 and both endpoints x = −3 and x = 3 are not included.

Similarly, the series ∑ d_n x^n has interval of convergence [−2, 2], so its radius is R2 = 2 and both endpoints x = −2 and x = 2 are included.

Now consider the combined series ∑ (c_n − d_n) x^n. This is a new power series whose coefficients are e_n = c_n − d_n.

Key idea: the radius (and thus the interval) of convergence of this new series is governed by the smallest radius among the original two, because within the overlap of convergence the difference of two convergent series is itself convergent termwise.

- For |x| < 2 (i.e., x in (−2, 2)), both original series converge absolutely (and thus converge), so their termwise difference ∑ (c_n − d_n) x^n converges.
- At x = 2: since ∑ c_n x^n converges at x = 2 (because 2 lies inside (−3, 3)) and ∑ d_n x^n also converges at x = 2 (since x = 2 is an endpoint included for the second series), their difference also converges at x = 2.
- At x = −2: similarly, both original series converge at x = −2 (−2 lies inside (−3, 3) for the first and is included for the second), so their difference converges at x = −2.

Therefore, the combined series converges at the endpoints x = −2 and x = 2 as well as on the open interval (−2, 2). This yields an inclusive interval of convergence: [−2, 2].

In summary, despite having the first series defined on (−3, 3) and the second on [−2, 2], the difference series inherits the overlap of convergence and includes the endpoints where both original series converge, giving the interval [−2, 2].

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