## anonymous 3 years ago Using cauchy's criterion |a_n+p - a_n|<epsilon prove that a_n= cos(x)/3 + cos(2*x)/3^2 + ..... + cos(n*x)/3^n is a cauchy sequence

1. anonymous

Using $|a_{n+p}-a _{n}|<$ Show that $a_{n}=\frac{ \cos x }{ 3 }+\frac{ \cos 2x }{ 3^{2} } + \frac{ \cos 3x }{ 3^{3} } + .... +\frac{ \cos nx }{ 3^{n} }$ is a cauchy sequence

2. anonymous

$|a_{n+p}-a _{n}|< \epsilon$

3. anonymous

since cosine is bounded above by 1 and below by -1 i believe this is the same as saying $\frac{1}{3^{n+p}}-\frac{1}{3^n}|<\epsilon$

4. anonymous

will that be enough ?:D

5. anonymous

should be although maybe we need to be a little careful

6. anonymous

the largest the absolute value can be is if one cosine is -1 and the other is 1 but the difference is bounded by 2

7. anonymous

so you can factor a 2 out of the whole thing, makes no difference in the proof

8. anonymous

by which i mean $|\frac{\cos((n+p)x)}{a^{n+p}}-\frac{\cos(nx)}{a^n}\leq 2|\frac{1}{3^{n+p}}-\frac{1}{3^n}|$

9. anonymous

$$a$$ should be $$3$$

10. anonymous

|dw:1353710521064:dw|

11. anonymous

Thank you both very much :D