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+⋯

The series is an illustration of a geometric series.

The series is given as:

So, we have:

-- first term

--- common ratio

The sum of n terms of a geometric series is:

The number of elements in the series is 4.

So, we have:

Hence, the sum of the elements in the series is

When the sum converges, we have:

So, we have:

The sum converges at S4.

So, we have:

Cancel out common factors

Divide both sides by -1

Add 1 to both sides

Divide by 16

Take 4th roots

Rewrite as:

Hence, the value of the variable is 0

Read more about geometric series at:

link

207/8 is the right

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!