You solve that inequality..surely you can do that, no?dwsmith said:$\displaystyle\sum_{n=0}^{\infty}\left(\frac{z}{z+1}\right)^n$$z\in\mathbb{C}$
By the ratio test,
$\displaystyle\left|\frac{z}{z+1}\right|<1$
I am stuck at this part. How do I find the z such that some is convergence?
AlexYoucis said:You solve that inequality..surely you can do that, no?
dwsmith said:Apparently I can't because I keep getting it wrong.
AlexYoucis said:Ok, so we need to solve $|z|<|z+1|$ or $x^2+y^2<(x+1)^2+y^2$ or $x^2<(x+1)^2$ or $x>\frac{-1}{2}$.
dwsmith said:How did you go from this $|z|<|z+1|$ to this $x^2+y^2<(x+1)^2+y^2$??
AlexYoucis said:Let $z=x+iy$.
dwsmith said:I did but I don't see what happened to all the i's.
AlexYoucis said:If $z=x+iy$ then $|z|=x^2+y^2$.
dwsmith said:Shouldn't it be square rooted?
The letter "z" is commonly used in mathematics to represent a complex number, which consists of a real part and an imaginary part. In the context of determining convergence, z represents a variable that can take on different values that affect the behavior of a given sequence or series. By varying the value of z, we can observe how the sequence or series converges or diverges.
The value of z can greatly impact the convergence of a series. For example, if the value of z is within the radius of convergence of a power series, the series will converge. However, if the value of z is outside the radius of convergence, the series will diverge. Additionally, the value of z can also affect the rate of convergence, with certain values leading to faster or slower convergence.
Yes, changing the value of z can significantly alter the convergence behavior of a series. For instance, a series that initially converges for a certain value of z may diverge for another value of z. This is why it is crucial to understand the role of z in determining convergence and to carefully choose its value when analyzing a series.
To determine the convergence of a series for a specific value of z, we can use various tests such as the Ratio Test, Root Test, or Alternating Series Test. These tests help us determine if the series converges or diverges for a given value of z. It is important to note that these tests may not always provide a definitive answer, and further analysis may be required.
No, there are other factors that can impact the convergence of a series, such as the form of the series and the magnitude of the terms. Additionally, the presence of other variables or parameters in the series may also affect its convergence behavior. It is essential to consider all these factors when determining the convergence of a series for a specific value of z.