Statement
Suppose that is a sequence of real- or complex-valued functions defined on a set, and that there is a sequence of positive numbers satisfying
for all and all . Suppose also that the series
is convergent.
Then the series
converges uniformly on .
Remark The result is often used in combination with the uniform limit theorem. Together they say that if, in addition to the above conditions, the set is a topological space and the functions are continuous on, then the series converges to a continuous function.
Read more about this topic: Weierstrass M-test
Famous quotes containing the word statement:
“Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
—Ralph Waldo Emerson (1803–1882)
“Children should know there are limits to family finances or they will confuse “we can’t afford that” with “they don’t want me to have it.” The first statement is a realistic and objective assessment of a situation, while the other carries an emotional message.”
—Jean Ross Peterson (20th century)
“Truth is used to vitalize a statement rather than devitalize it. Truth implies more than a simple statement of fact. “I don’t have any whisky,” may be a fact but it is not a truth.”
—William Burroughs (b. 1914)