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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_nf_n(x),

2. suma_n is convergent,

3. f_n(x) is a monotonic decreasing sequence (i.e., f_(n+1)(x)<=f_n(x)) for all n, and

4. f_n(x) is bounded in some region (i.e., 0<=f_n(x)<=M for all x in [a,b])

then, for all x in [a,b], the series sumu_n(x) converges uniformly.

See also

Convergence Tests, Convergent Series, Uniform Convergence

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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.Jeffreys, H. and Jeffreys, B. S. "Abel's Lemma" and "Abel's Test." §1.1153-1.1154 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 41-42, 1988.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, p. 17, 1990.

Referenced on Wolfram|Alpha

Abel's Uniform Convergence Test

Cite this as:

Weisstein, Eric W. "Abel's Uniform Convergence Test." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AbelsUniformConvergenceTest.html

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