# Closed forms of summations and reprospective relations

• Jan 14th 2012, 11:40 PM
grainofsand
Closed forms of summations and reprospective relations
Hello and have a nice week !
Im asking your help with the following exercices (tried but failed).

So the first one
Find the closed form of the double summation
S(from j=1 to n) S(from i=0 to infinity) j^5/3 (1-1/2j^1/3)^I

The second one
(a) T(n)=T(n/4)+log4_n and T(1)=0 fond the closed form .
(b) A flower can live for 2 years and reproduct once a year.Specifically it reproducts during its first year of life . Find a reprospective relation that describes the number of flowers in time n and then find a closed form if we know that in time 0 there is only one flower.

Thanks in advance !!!!
• Jan 15th 2012, 01:31 AM
CaptainBlack
Re: Closed forms of summations and reprospective relations
Quote:

Originally Posted by grainofsand

So the first one
Find the closed form of the double summation
S(from j=1 to n) S(from i=0 to infinity) j^5/3 (1-1/2j^1/3)^I

Put in sufficient brackets to make your expressions unambiguous!, As it stands there is no closed form as the inner sum is divergent.

So let us assume you mean:

$S(n)=\sum_{j=1}^n \sum_{i=0}^{\infty} j^{5/3}\left(1-\frac{1}{2j^{1/3}}\right)^i$

Write it as:

$S(n)=\sum_{j=1}^n j^{5/3} \sum_{i=0}^{\infty} \left(1-\frac{1}{2j^{1/3}}\right)^i$

Now the inner sum is an infinite geometric series with sum:

$S_{inner}(j)= \frac{1}{1-(1-1/(2j^{1/3}))}=2j^{1/3}$

...

CB
• Jan 16th 2012, 01:48 AM
grainofsand
Re: Closed forms of summations and reprospective relations
Quote:

Originally Posted by CaptainBlack
Put in sufficient brackets to make your expressions unambiguous!, As it stands there is no closed form as the inner sum is divergent.

So let us assume you mean:

$S(n)=\sum_{j=1}^n \sum_{i=0}^{\infty} j^{5/3}\left(1-\frac{1}{2j^{1/3}}\right)^i$

Write it as:

$S(n)=\sum_{j=1}^n j^{5/3} \sum_{i=0}^{\infty} \left(1-\frac{1}{2j^{1/3}}\right)^i$

Now the inner sum is an infinite geometric series with sum:

$S_{inner}(j)= \frac{1}{1-(1-1/(2j^{1/3}))}=2j^{1/3}$

...

CB

really thanks for the help !(Rofl)(Rofl)(Rofl)(Rofl)(Rofl)(Rofl)
i solved the first one !!!!!!!
But what about the other one with the 2 parts ????
And again thanks in advance !
for the second one part a i found sth but i dont think it is right (cause i dont even use the t(1)=0).
T(n)=O((log4_n)*n^(log4_2) How can i find the closed form of the reprospective relation ?
and what about the part b ?