请教几道GR0568的题目，谢谢！

mccrimsonUID: 2636067

1  If z=e^(2*pi*i/5), then 1+z+z^2+z^3+5z^4+4z^5+4z^6+4z^7+4z^8+5z^9 = ?
我感觉答案应该是5z^4  不知道为什么选 -5 e^(3*pi*i/5) ?

2. if [x] denotes the greatests integer not exceeding x, then 从0到无穷 对 [x]e^(-x) dx 积分 = ?
答案是1/(e-1) 不知道为什么

3. Let K be a nonempty subset of R^n, where n>1. If every continuous real-valued function defined on K is bounded, then k is compact. 这个是为什么？

4. For each positive integer n, let f_n be the function defined on theinterval [0,1] by f_n(x)=x^n /(1+x^n) 那么为啥 lim_{n->inf} 从0到1对f_n(x)积分 = 从0到1 对 lim_{n->inf} f_n(x) 积分？

谢谢

soarerzUID: 2721318

1. 1+z+z^2+z^3+z^4 + 4(z^4+z^5+z^6+z^7+z^8) +5z^9 = 5z^9 = 5z^4 = 5e^(8pi i/5) = -5 e^(3 pi i/5) (because e^pi i = -1)
2. integral = int_1^2 1*e^(-x) dx + int 2^3 2*e^(-x)dx + int 3^4 3*e^(-x)dx + ...
sum this series.
3. if K is not compact, it's either not bounded or not closed. If it is not bounded you can just pick a function that diverges to infinity, say f(x) = ||x||. If it is not closed but bounded, say there is a limitpoint x not in the set, then you can imagine a function that has a pole at x. That function is continuous but unbounded on K.
4. dominated convergence theorem.

mccrimsonUID: 2636067

明白了，谢谢soarerz!