## graydarl 2 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. graydarl

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. graydarl

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

3. satellite73

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. graydarl

will that be enough ?:D

5. satellite73

should be although maybe we need to be a little careful

6. satellite73

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. satellite73

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

8. satellite73

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. satellite73

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

10. mahmit2012

|dw:1353710521064:dw|

11. graydarl

Thank you both very much :D