8.3. Series and Power Series

Example 8.3.9 (c): Power Series Center

Consider the series 2-n (x+2)n Why can you not re-center this series at c = 0?

This series has center of convergence c = -2 and, as you can confirm, radius r = 2. Thus, it converges for

|x + 2| < 2

In particular, the series diverges for x = 0, one of the endpoints of this interval.

But if we could find a series centered at c = 0 it would, in particular, converge at its center. But since we already know the original series does not converge for x = 0 we can not rewrite it with c=0 as center.

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