Use the properties of geometric series to find the sum of the series. for what values of the variable does the series converge to this sum? 5+(−10)z+20z2+(−40)z3+⋯

[tex]5\times(-2)^0=5[/tex]
[tex]5\times(-2)^1=-10[/tex]
[tex]5\times(-2)^2=20[/tex]
[tex]5\times(-2)^3=-40[/tex]
...

There's enough of a pattern here to discern the series to be

[tex]\displaystyle\sum_{k=0}^\infty 5(-2z)^k[/tex]

which converges as long as [tex]|-2z|=2|z|<1[/tex], or [tex]|z|<\dfrac12[/tex]. Under this condition, the series would converge to the function

[tex]f(z)=\dfrac5{1+2z}[/tex]

Answer Link

MrRoyal MrRoyal · Answer 2 · 2021-11-03T17:26:44+01:00

The series is an illustration of a geometric series.

The sum of the elements in the series is [tex]\mathbf{-\frac{5(16z^4 - 1)}{2z + 1}}[/tex]
The variable is 0, when the sum converges at [tex]\mathbf{-\frac{5(16z^4 - 1)}{2z + 1}}[/tex]

The series is given as: [tex]\mathbf{5+(-10)z+20z^2+(-40)z^3+...}[/tex]

So, we have:

[tex]\mathbf{a = 5}[/tex] -- first term

[tex]\mathbf{r= -2z}[/tex] --- common ratio

The sum of n terms of a geometric series is:

[tex]\mathbf{S_n = \frac{a(r^n - 1)}{r - 1}}[/tex]

The number of elements in the series is 4.

So, we have:

[tex]\mathbf{S_4 = \frac{5((-2z)^4 - 1)}{-2z - 1}}[/tex]

[tex]\mathbf{S_4 = \frac{5(16z^4 - 1)}{-2z - 1}}[/tex]

[tex]\mathbf{S_4 = -\frac{5(16z^4 - 1)}{2z + 1}}[/tex]

Hence, the sum of the elements in the series is [tex]\mathbf{-\frac{5(16z^4 - 1)}{2z + 1}}[/tex]

When the sum converges, we have:

[tex]\mathbf{S_{\infty} = \frac{a}{1 -r}}[/tex]

So, we have:

[tex]\mathbf{S_{\infty} = \frac{5}{1 +2z}}[/tex]

The sum converges at S4.

So, we have:

[tex]\mathbf{\frac{5}{1 +2z} = -\frac{5(16z^4 - 1)}{2z + 1}}[/tex]

Cancel out common factors

[tex]\mathbf{1 = -(16z^4 - 1)}[/tex]

Divide both sides by -1

[tex]\mathbf{-1 = 16z^4 - 1}[/tex]

Add 1 to both sides

[tex]\mathbf{0 = 16z^4 }[/tex]

Divide by 16

[tex]\mathbf{0 = z^4 }[/tex]

Take 4th roots

[tex]\mathbf{0 = z}[/tex]

Rewrite as:

[tex]\mathbf{z = 0}[/tex]

Hence, the value of the variable is 0

Read more about geometric series at:

https://brainly.com/question/4617980