Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am trying to evaluate an expression which is the arithmetic mean of first $N$ partial sums of a geometric progression.It is given as below.

$\frac{1}{N}\sum\limits_{k=0}^{N-1}(N-k)z^k$

Please suggest me some hints or ideas to proceed.

share|improve this question
add comment

1 Answer

up vote 4 down vote accepted

More generally, you can evaluate

$$\sum_{k=0}^{N-1}P(k)z^k$$

for any polynomial $P$ by using

$$(z\frac{\mathrm d}{\mathrm d z}) z^k=kz^k\;.$$

Thus, you can replace $k$ by $D:=z\frac{\mathrm d}{\mathrm d z}$ in $P\,$:

$$ \begin{eqnarray} \frac{1}{N}\sum_{k=0}^{N-1}(N-k)z^k &=& \sum_{k=0}^{N-1}\left(1-\frac{k}{N}\right)z^k \\ &=& \sum_{k=0}^{N-1}\left(1-\frac{D}{N}\right)z^k \\ &=& \left(1-\frac{D}{N}\right)\sum_{k=0}^{N-1}z^k \\ &=& \left(1-\frac{D}{N}\right)\frac{z^N-1}{z-1}\;. \end{eqnarray} $$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.