\[ \sum_{k=0}^{100} 3^{-k}Cos(\frac{k \pi}{2}) \], \[ \sum_{k=0}^{200} 3^{-k}Sin(\frac{k \pi}{2}) \]
I have to prove that one of these sums are about 3 times as larger as the other.
Using the unit circle I figured out this pattern for Cosine, (1,0,-1,0) and for Sin(0,1,0,-1)
this means we can ignore 50 summations for cos and 100 for sin since they will be 0.
For cosine I wrote out the first 4 summations, [MATH]1+0-3^{-2}+0+3^{-4}...[/MATH] Removing the 0s and moving the 1 to the front as well as realizing we will get the same summation of [MATH]-3^{-2}+3^{-4}[/MATH] every time except it will be [MATH]\frac {1}{3^2}[/MATH] Smaller, I rewrite the sum as:
\[ 1+\sum_{k=0}^{50} (\frac{-3^{-2}+3^{-4}}{3^{2k}})\]
I then do the same for Sin but It doesn´t feel like I´ve proven anything. Sure after 2-3 summations I get the correct answers but is this really what was intended? That I simplify or rewrite the summations and take the sum of 2-3 iterations and then compare? From this I get the cos sum to be about 0.9 and the Sine one to 0.3 exact.
Sin was rewritten to \[ \sum_{k=0}^{100} (\frac{3^{-1}-3^{-3}}{3^{4k}})\] Using the same logic.
Is my method correct or did I completely miss the "proving" part of the question? As I said, feels like I just rewrote it and although it shows me if i take 2-3 sums of each what the answer will be, I don´t know how I can argue that it is the finite value the sums will get.
I have to prove that one of these sums are about 3 times as larger as the other.
Using the unit circle I figured out this pattern for Cosine, (1,0,-1,0) and for Sin(0,1,0,-1)
this means we can ignore 50 summations for cos and 100 for sin since they will be 0.
For cosine I wrote out the first 4 summations, [MATH]1+0-3^{-2}+0+3^{-4}...[/MATH] Removing the 0s and moving the 1 to the front as well as realizing we will get the same summation of [MATH]-3^{-2}+3^{-4}[/MATH] every time except it will be [MATH]\frac {1}{3^2}[/MATH] Smaller, I rewrite the sum as:
\[ 1+\sum_{k=0}^{50} (\frac{-3^{-2}+3^{-4}}{3^{2k}})\]
I then do the same for Sin but It doesn´t feel like I´ve proven anything. Sure after 2-3 summations I get the correct answers but is this really what was intended? That I simplify or rewrite the summations and take the sum of 2-3 iterations and then compare? From this I get the cos sum to be about 0.9 and the Sine one to 0.3 exact.
Sin was rewritten to \[ \sum_{k=0}^{100} (\frac{3^{-1}-3^{-3}}{3^{4k}})\] Using the same logic.
Is my method correct or did I completely miss the "proving" part of the question? As I said, feels like I just rewrote it and although it shows me if i take 2-3 sums of each what the answer will be, I don´t know how I can argue that it is the finite value the sums will get.