What Property Connects the Steps in a Geometric Series Calculation?

[nacho-man]

Hi, just wanted to know what property/?? was used to get from the first red-box to the second one.

It looks like it has to do with the geometric sum, but my series/sequence is verrrrrry rusty.

any help appreciated, I'd like to find the general property so I can apply this rule to different examples. I can see that the bounds of the sum has altered the outcome also.

IF anyone could explain this to me, or link me to some relevant theory I would be ecstatic.many thanks in advance.

(see attached image)edit: my brain is fried, there are tables which give these direct results! awesome.
just wondering though, is there a way to solve for z-transforms without having to consult a table?
Although this last question is probably reserved for a different sub-board.

thanks anyway.!

 

[Dethrone]

That is the sum of a geometric series, consider:

We'll represent a geometric series by $S_n=\sum ar^{n-1}$

$$S_n=a+ar+ar^2+...+ar^{n-1}$$

Multiply both sides by $r$:

$$rS_n=ar+ar^2+...+ar^{n-1}+ar^n$$

Subtract from $S_n$ so that most of the terms cancel:

$$S_n - rS_n=a-ar^n$$
$$(1-r)S_n=a(1-r^n)$$
$$S_n = \frac{a(1-r^n)}{1-r}$$

$a$ is the first term, which in your case, is $1$.
$r$ is the common ratio which is what we're multiplying "$n$" times: $az^{-1}$

Applying $S_n$, we get from the first red box to the second :D I don't think I'll be able to answer your other questions though, I've only just started learning series.

 

[I like Serena]

edit: my brain is fried, there are tables which give these direct results! awesome.
just wondering though, is there a way to solve for z-transforms without having to consult a table?

Hi nacho,

Evaluating a z-transform without tables is exactly what is done here.
It depends on evaluating a series.

Tables or math software make your life much easier for complicated z-transforms though.

 

[nacho-man]

wonderful, thanks heaps guys.
A revision session is long overdue haha!

 

[CellCoree]

Hello,

The property that was used to get from the first red-box to the second one is called the geometric series/sum property. This property states that for a geometric sequence with common ratio r, the sum of the first n terms can be calculated using the formula Sn = a(1-r^n)/(1-r), where a is the first term and n is the number of terms.

In this case, it looks like the first red-box has a geometric sequence with a common ratio of 2, and the second red-box has a geometric sequence with a common ratio of 4. This means that the bounds of the sum have been altered by changing the number of terms (n) and the common ratio (r).

To apply this rule to different examples, you can use the formula mentioned above and plug in the appropriate values for a, n, and r. As for solving for z-transforms without consulting a table, you can use the properties and rules of z-transforms to manipulate the equations and find the answer. However, consulting a table can be helpful for more complex problems.

I hope this helps and good luck with your studies!