1. The problem statement, all variables and given/known data I need to show that both sin(x) and cos(x) are absolutely convergent. Here's my work so far, Theorem: ℯix = cos(x) + i*sin(x) (1) Proof: This proof will be one using the power series. Note: i = i, i^2 = -1 i^3 = -i, i^4 = 1, i^5 = i i^6 = -1 i^7 = -i i^8 = 1, etc. for all positive integers and that the infinite series for sin(x) and cos(x), from the Taylor Series, are as follows: sin(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - x^10/10! + … cos(x) = x/1! - x^3/3! + x^5/5! - x^7/7! + x^9/9! - x^11/11! + ... Using the power series definition for ℯz, we can say that the power series could be written as: ℯ^(z) = 1 + z/1! + z^2/2! + z^3/3! + z^4/4! + … + The infinite sum, starting at 0, of z^n/n! Using equation (2), Euler did, and we will using ix for our exponent, and treat the imaginary i product with x as a real number. This gives: ℯix = 1/0! + ix/1! + (ix)2/2! + (ix)3/3! + (ix)4/4! + … + The infinite sum, starting at 0, of z^n/n! This gives the result of: ℯ^(ix) = 1 + ix/1! -x^2/2! - ix^3/3! + x^4/4! + ix^5/5! -x^6/6! - ix^7/7! + x^8/8! + ... We can separate the terms to be such that: ℯ^(ix) = (1 - x^2/2! + x^4/4! -x^6/6! + x^8/8! - …) + (ix/1! - ix^3/3! + ix^5/5! - ix^7/7! + ... ) If we separate the term i out of the terms containing i we can see ℯ^(ix) = (1 - x^2/2! + x^4/4! -x^6/6! + x^8/8! - …) + i(x/1! - x^3/3! + x^5/5! - x^7/7! + ... ) Interestingly, we can see that the series that has i as a factor and the other that is not, are both infinite series. The former being sin(x) and the latter being cos(x). The only thing we need to prove is that the two series are absolutely convergent. From Calculus we know that there is a theorem to test absolute convergence: By looking at the series ℯx we can determine if cos(x) converges absolutely. To do that we have the following rule. ... A series is absolutely convergent if the sum of the absolute values of the terms is also convergent. Which means that we can express ℯix as: ℯix = cos(x) + i*sin(x) Which is what we were trying to prove. 2. Relevant equations All forms of convergence tests (i.e. Ratio Test, Integral Test, Alternating Series test etc.) 3. The attempt at a solution I have tried to use all three tests, but have failed. Any help is very appreciated. I know that e^(x) is absolutely convergent I'm just trying to make the connection.