This is something that comes up when I want to determine whether the sequence of functions {f_n} converge uniformly to f:
Suppose f_n(x) = sqrt(x^2 + 1/n^2), so f(x) = x.
Then, according to Spivak, f(x) - f_n(x) = sqrt(x^2) - sqrt(x^2 + 1/n^2) = 1/(2n^2*sqrt(ε)) for some ε such that x^2 < ε...
If n>m>1, find upper and lower bounds for n!/m!
Answer: upper bound = {[(n+1)^(n+1)]/[(m+1)^(m+1)]}*e^(-(n-m))
lower bound = [(n^n)/(m^m)]*e^(-(n-m))
This is from a chapter on finding bounds for sum of series. Can someone please explain how to arrive at the answer? Thanks!
I wanted to post this in the homework forum, but there's only pre-calc for math.
Question: Show that i*r^i-(i-1)*r^(i-1) = r^(i-1)-(1-r)i*r^i-1. Use this result to find the sum of i*r^(i-1) from i=1 to i=n.
I've done the first part of this question, but need some help with the second. Thanks!