- #1
thanksie037
- 18
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Homework Statement
Find the sum of the series s(x) = 1 +cos(x)+ (cos2x)/2!+(cos3x)/3!...
thanksie037 said:would x= cosnx? is there a property for cosine where (cos x)^n=cos nx ?
thanksie037 said:Do you mean:
x^ni=cos nx + sin nx?
thanksie037 said:sorry i was using that first one. i was confused as to what you meant with cos nx.
using the second one makes a lot more sense. are you saying i should:
cos n(theta) = (e^ni(theta) +e^-ni(theta))/2
thanksie037 said:and how do i even find the sum of this series. i only know how to find a sum of a geometric series...i don't know how to deal with the factorial
thanksie037 said:do the series n/(n+1) and n^2/(n^2+1)converge or diverge?
gabbagabbahey said:Huh?! Why on Earth would you think that were true?
Have you not seen the formulas [tex]e^{i\theta}=\cos\theta+i\sin\theta[/tex] and [tex]\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}[/tex] before?
The formula for finding the sum of a series involving cosine is S = a + a cos(x) + a cos(2x) + a cos(3x) + ... + a cos(nx), where a is the amplitude and x is the angle in radians.
A series involving cosine converges if the limit of the absolute value of the terms as n approaches infinity is equal to 0. If the limit is any other value, the series diverges.
Yes, the sum of a series involving cosine can be negative. This depends on the values of a and x in the formula. If a is negative, the sum will also be negative.
Yes, there are a few special cases when finding the sum of a series involving cosine. One is when x is equal to 0, in which case the sum will simply be (n+1)a. Another is when a is equal to 0, in which case the sum will be 0 regardless of the value of x.
Yes, the sum of a series involving cosine can be simplified in some cases. For example, if the series involves cos(nx), it can be simplified using the double angle formula for cosine. Additionally, if the series involves cos(x), it can be simplified using the power series expansion for cosine.