Homework Statement
I need to prove that
\frac{\frac{1}{2} cot20^{o}-cos10^{o}}{\frac{1}{2}+sin10^{o}}=\frac{\sqrt{3}}{3}The Attempt at a Solution
I try to do it by this way
\frac{\frac{1}{2}...
Homework Statement
It is given that (3+5\sqrt{2})^n=(5+3\sqrt{2})^m, wgere n, m are natural numbers.Homework Equations
Show that (3-5\sqrt{2})^n=(5-3\sqrt{2})^m
Of course, that's my careless mistake. It is obvious that from the definition of infimum we have that for every \epsilon>0 exists a natural number n_{0} such that \frac{a_{n_{0}}}{n_{0}}<a+\epsilon.
I found the solution of my problem.
Denote a=inf\{\frac{a_{n}}{n}\}. a is finite, because the set \{\frac{a_{n}}{n}\} is bounded (a\ge 0). For every \epsilon>0 exists a natural number n_{0}, such that \frac{a_{n_{0}}}{n_{0}}<\epsilon.
Let us fix the number n_{0}.
Every n\in {N} we can...
I have already understood that the sequence \frac{a_{n}}{n} can not be non-increasing. I have found some way to solve this problem. I have already shown that the sequence a_{n} is bounded, so I can take from that sequence a sub-sequence, say a_{n_k}, having limit a. If you can help me to show...
It is obvious that the limit exists if \frac{a_n}{n} is nonincreasing, but I can't prove that sequence \frac{a_n}{n} is nonincreasing... The problem is to show the last one :)
Homework Statement
It is given the sequence a_{n}, where a_{n}>0 and a_{n+m}\leq a_{n}+a_{m}Homework Equations
Prove that the limit lim_{n\rightarrow \infty} \frac{a_{n}}{n} exists.The Attempt at a Solution
I have shown that a_{n}\leq n a_{1}. From the last inequality I obtain that...
Let P(n)=\int\frac{1-\cos{nx}}{1-\cos{x}}dx
P(n+1)-P(n)=\int\frac{\cos{(n+1)x}-\cos{nx}}{1- \cos{x}}dx=\int\frac{2\sin{(n+\frac{1}{2})x} \sin{\frac{x}{2}}}{2\sin^{2}{\frac{x}{2}}}dx=\int \frac{\sin{(n+\frac{1}{2})x} }{\sin{\frac{x}{2}}}dx
(thr limits of integral are 0 and pi)
The last Integral...