Help with Proof: Sum of Cosines Up to n Terms

  • Context: MHB 
  • Thread starter Thread starter shen07
  • Start date Start date
  • Tags Tags
    Proof
Click For Summary
SUMMARY

The discussion centers on the mathematical proof for the sum of cosines up to n terms, specifically represented by the equation $$\sum_{k=0}^{k=n}(nCk * cos(kx)) = cos(nx/2)*(2cos(x/2))^n$$. Participants utilized the substitution $$z=exp(ix)$$ to derive the result, leading to the expression $$(1+z)^n = 2^n cos^n (x/2) cos (nx/2)$$. The final answer is confirmed as the real part of the derived expression, showcasing a clear method for solving the problem.

PREREQUISITES
  • Understanding of complex numbers and Euler's formula
  • Familiarity with binomial coefficients, denoted as $$nCk$$
  • Knowledge of trigonometric identities, particularly cosine
  • Experience with mathematical proofs and manipulation of series
NEXT STEPS
  • Study the derivation of Euler's formula and its applications in trigonometry
  • Learn about binomial expansions and their relevance in combinatorial mathematics
  • Explore the properties of cosine functions and their sums
  • Investigate the use of complex numbers in solving real-valued problems
USEFUL FOR

Mathematicians, physics students, and anyone interested in advanced trigonometric identities and their proofs will benefit from this discussion.

shen07
Messages
54
Reaction score
0
$$\sum_{k=0}^{k=n}(nCk * cos(kx)) = cos(nx/2)*(2cos(x/2))^n$$
 
Physics news on Phys.org
I also know that i have to use

$$
z=exp(ix)
$$
$$
(1+z)^n = 2^n cos^n (x/2) cos (nx/2) $$
 
Got the answer..
 
shen07 said:
Got the answer..

Hi shen07 (Wave),

Welcome to MHB! Sorry we couldn't help you quickly enough this time. I'm sure that in the future you'll find guidance with something you are stuck on. Care to share your answer so others may see it?

Jameson
 
The answer could be obtained by choosing the real part after substituting $z=e^{ix}$

Hence we have

$$\Large (1+e^{ix})^n= e^{\frac{ixn}{2}}\left(e^{\frac{-ix}{2}}+e^{\frac{ix}{2}}\right)^n= 2^n e^{\frac{ixn}{2}}\cos^n \left(\frac{x}{2}\right)$$

Clearly the answer is the real part of the above expression .
 

Similar threads

Undergrad Regarding Wirtinger's inequality
  • · Replies 4 ·
Replies
4
Views
3K
Calculus ## G(y, z)=z\frac{\partial}{\partial z}F(y, z)-F(y, z) ##?
  • · Replies 6 ·
Replies
6
Views
2K
MHB Prove that series converges uniformly
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Question about intersection of coordinate rings
  • · Replies 1 ·
Replies
1
Views
1K
MHB The proof of the infinite geometric sum
  • · Replies 1 ·
Replies
1
Views
1K
MHB Checking Proof of Theorem 6.2.8 Part (ii)
  • · Replies 2 ·
Replies
2
Views
3K
MHB Outer measure of a closed interval .... Axler, Result 2.14 ....
  • · Replies 2 ·
Replies
2
Views
1K
MHB Weierstrass Function: Continuous and Bounded on $\mathbb{R}$
  • · Replies 38 ·
2
Replies
38
Views
5K
How should I show the following by using the signum function?
  • · Replies 14 ·
Replies
14
Views
2K
MHB Outer measure .... Axler, Result 2.8 ....
  • · Replies 2 ·
Replies
2
Views
3K