Abel's Uniform Convergence Test

Let $\{u_n(x)\}$ be a Sequence of functions. If

1. $u_n(x)$ can be written $u_n(x) = a_n f_n(x)$,

2. $\sum a_n$ is Convergent,

3. $f_n(x)$ is a Monotonic Decreasing Sequence (i.e., $f_{n+1}(x)\leq f_n(x)$) for all $n$, and

4. $f_n(x)$ is Bounded in some region (i.e., $0 \leq f_n(x) \leq M$ for all $x \in [a,b]$)

then, for all $x \in [a,b]$, the Series $\sum u_n(x)$ Converges Uniformly.

See also Convergence Tests

References

Bromwich, T. J. I'a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 59, 1991.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, p. 17, 1990.