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# Uniform convergence watch

1. I'm really struggling with this question:

h:[0,1] -> R is continuous
Prove that t(x) = xnh(xn) is uniformly convergent on [0,s] where 0<s<1

I have the definition of h being continuous but after this I am pretty clueless about how to tackle this problem. I could use the Weierstrass M-test. I know the series xn converges uniformly on this interval as xn < sn but I don't know how to use the fact that h is continuous to find a sequence of real numbers that xnh(xn) is always less than.
2. h is continuous on a closed interval implies.....
3. Oh yeah, that it is bounded, so h(xn) < M say.

So xnh(xn) < Msn.
4. I'm really struggling with another question now:

0<p<1
Suppose p(p-1)...(p-k+1)(-1)k/k(k-1)...1 is convergent.
Show that p(p-1)...(p-k+1)(x)k/k(k-1)...1 is uniformly convergent on [-1,0]

I have shown that p(p-1)...(p-k+1)(-1)k/k(k-1)...1 < 0 for k=1,2,3,...
p(p-1)...(p-k+1)(-1)k/k(k-1)...1 = L (< 0) as it converges to a limit.
|(-1)krk| rk for r<1 and -1<x0
However, I do not know how to tackle the case when x=-1.

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Updated: February 14, 2010
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