# I need solution

air-man001

hello every one

I have this summation in my book

infinity
$$\sum$$ k*z-k
k=0

my solution:

infinity
$$\sum$$ k*z-k = z-1+2z-2+3z-3+............
k=0

please i want the closed form (step-by-step)

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Homework Helper
Gold Member
Here's a hint. The sum might as well start at k = 1. Let z = 1/x

$$\sum_{k=1}^n kx^k = x\sum_{k=1}^n kx^{k-1}$$

What is inside the sum on the right you should recognize as a derivative of a series that you should be able to sum. Can you take it from there?

air-man001
LCKurtz , i can't understand your post.

I know the closed form of geometric series =$$\frac{FirstTerm} {1 - CommonRatio}$$

CommonRatio = termn / termn-1

but in my problem , the 'CommonRatio' isn't fixed

term4/term3 not equal to term3/term2

Mentor
I know the closed form of geometric series =$$\frac{FirstTerm} {1 - CommonRatio}$$

CommonRatio = termn / termn-1

but in my problem , the 'CommonRatio' isn't fixed

term4/term3 not equal to term3/term2
Which should suggest to you that your series isn't a geometric series and you're heading down the wrong path. LCKurtz has given you an alternative direction, so see what you can do with that.

Homework Helper
Gold Member
Here's a hint. The sum might as well start at k = 1. Let z = 1/x

$$\sum_{k=1}^n kx^k = x\sum_{k=1}^n kx^{k-1}$$

What is inside the sum on the right you should recognize as a derivative of a series that you should be able to sum. Can you take it from there?

LCKurtz , i can't understand your post.

I know the closed form of geometric series =$$\frac{FirstTerm} {1 - CommonRatio}$$

CommonRatio = termn / termn-1

but in my problem , the 'CommonRatio' isn't fixed

term4/term3 not equal to term3/term2

Do you understand the part of my hint about

$$\sum_{k=1}^n kx^{k-1}$$

being the derivative of something?

air-man001
LCKurtz

it is derivative of xk
right?

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air-man001
mr Mark44

I am studying "z-transform" in Digital Signal Processing,

this is not homework, i studying for final Exam tomorrow

Last edited:
Homework Helper
Gold Member
Do you understand the part of my hint about

$$\sum_{k=1}^n kx^{k-1}$$

being the derivative of something?

LCKurtz

it is derivative of xk
right?

Yes, so you have a sum of derivatives. That is the same as the derivative of the sum and you should be able to calculate that sum and take its derivative.

air-man001
thank you mr.LCKurtz

I will try it now

air-man001
i can't do it!!!!!

Homework Helper
Gold Member
You have

$$\sum_{k=1}^n kx^k = x\sum_{k=1}^n kx^{k-1}= x\frac{d}{dx} \left (\sum_{k=1}^n x^k\right )$$

You know the formula for the sum of the geometric series. Use it and do the derivative.

air-man001
I'm sorry
i can't do it!!!!!

Homework Helper
Gold Member
I'm sorry
i can't do it!!!!!

Get out your algebra book and look up the formula for the sum of the first n terms of a geometric series and use it to evaluate the sum.

Then take its derivative. I'm not going to do it for you.

air-man001
Get out your algebra book and look up the formula for the sum of the first n terms of a geometric series and use it to evaluate the sum.

Then take its derivative. I'm not going to do it for you.

Firstly..Thank you for help me.

secondly.. this is not homework ... the subject in Digital signal processing (finding the z-transform) only.

air-man001
i will save the issue rather than understanding

Mentor
mr Mark44

I am studying "z-transform" in Digital Signal Processing,

this is not homework, i studying for final Exam tomorrow
Whether it's homework of exam review, we're not going to do your work for you. We'll help you do it and steer you in the right direction if/when you make a mistake, but you have to put in some of the effort.