MathCS.org - Real Analysis: Example 8.3.4 (b): Function Series Examples

Example 8.3.4 (b): Function Series Examples

On which interval does the series f(x) = 3²ⁿxⁿ represent a continuous function?

Back

Let's compute the sup-norm:

|| 3²ⁿxⁿ ||_{[-r, r]} = (9r)ⁿ

The numeric series (9r)ⁿ is finite iff 9r < 1 or equivalently r < 1/9.

Thus, our series converges absolutely and uniformly to a continuous function on every closed subset of (-1/9, 1/9).

Next | Previous | Glossary | Map | Discussion