Infinite series of complex numbers.

C and S, in terms of x. He has rewritten the series in terms of z, where z represents cosx + jsinx, and has found the sum to infinity of C + jS to be 3/(3-z). However, he is now stuck and is looking for assistance in breaking this answer into real and imaginary parts. In summary, Carl is seeking help with finding the sum to infinity of two series in terms of x, and is looking for assistance with breaking his current answer into real and imaginary parts.
  • #1
Nevermore
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I have the two series:
C = 1 + (1/3)cosx + (1/9)cos2x + (1/27)cos3x ... (1/3^n)cosnx
S = (1/3) sinx + (1/9)sin2x ... (1/3^n)sinnx

I have to express, in terms of x, the sum to infinity of these two series.

Here's what I've done so far:
Let z represent cosx + jsinx
C + jS = z^0 + (1/3)Z^1 ... (1/3)^n(Z^n)
This is a GP with first term 1, common ratio (1/3)Z.
Sum to infinity of C + jS = a/(1-r) = 3/(3-Z)

I can't seem to find where to go from there. Can anyone help?

Thankyou.
 
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  • #2
Write z=x+iy and break your answer, 3/(3-z) into real and imaginary parts.

Carl
 

1. What is an infinite series of complex numbers?

An infinite series of complex numbers is a sequence of complex numbers that goes on forever. It is typically denoted by the sum of the terms, with each term in the series being added to the previous term.

2. What is the difference between a finite series and an infinite series of complex numbers?

A finite series has a limited number of terms, while an infinite series goes on forever. Additionally, the terms in an infinite series of complex numbers are complex numbers, while the terms in a finite series can be any type of number.

3. How do you determine convergence or divergence of an infinite series of complex numbers?

The convergence or divergence of an infinite series of complex numbers can be determined using various tests such as the ratio test, root test, or comparison test. These tests involve examining the behavior of the terms in the series and determining if they approach a finite limit or continue to increase infinitely.

4. Can an infinite series of complex numbers converge to a complex number?

Yes, an infinite series of complex numbers can converge to a complex number. This is known as a convergent series, and it means that the sum of the terms in the series approaches a finite complex number as the number of terms increases.

5. What are some real-world applications of infinite series of complex numbers?

Infinite series of complex numbers have various applications in fields such as physics, engineering, and finance. They can be used to model and analyze complex electrical circuits, calculate the value of financial investments, and study the behavior of physical phenomena such as sound waves and electromagnetic fields.

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