Archives for Pure Math - Page 2
Let R > 1 and let f be analytic on IzI < R except at z = 1, where f has a simple pole. If M n=O is the Maclaurin series for f, show that hn,+,M alL exists.
To show that the limit limn→∞nan\lim_{n \to \infty} n a_nlimn→∞nan exists for the Maclaurin series f(z)=∑n=0∞anznf(z) = \sum_{n=0}^{\infty} a_n z^nf(z)=∑n=0∞anzn of the function fff, which is analytic in ∣z∣<R|z| <…
An accurate map of California is spread out flat on a table in Evans Hall, in Berkeley. Prove that there is exactly one point on the map lying directly over the point it represents.
To prove that there is exactly one point on the map of California that lies directly over the point it represents, we can use a combination of the Brouwer Fixed-Point…
Let f : R + R be continuous, with 00 s_, If(x)l dx < o. Show that there is a sequence (x,,) such that x, -+ 00, x, f (x,) 4 0, and x, f(-xc,) 4 0 as n -+ o.
Let f : R + R be continuous, with 00 s_, If(x)l dx < o. Show that there is a sequence (x,,) such that x, -+ 00, x, f (x,)…
Given conditions: lim 𝑡 → 𝑎 𝑔 ( 𝑡 ) = 𝑏 lim t→a g(t)=b lim 𝑥 → 𝑏 𝑓 ( 𝑥 ) = 𝑐 lim x→b f(x)=c To prove: lim 𝑡 → 𝑎 𝑓 ( 𝑔 ( 𝑡 ) ) = 𝑐 lim t→a f(g(t))=c
Given conditions: limt→ag(t)=b\lim_{t \to a} g(t) = blimt→ag(t)=b limx→bf(x)=c\lim_{x \to b} f(x) = climx→bf(x)=c To prove: limt→af(g(t))=c\lim_{t \to a} f(g(t)) = climt→af(g(t))=c Using the definition of the limit: limt→ag(t)=b\lim_{t \to…