Need help with euler,s transform of series

In summary, the conversation is about finding help with using Euler's transform to improve the convergence of a series. The series is related to the sum of all primes for the function z**(x), and the question is whether or not Euler's transform can be applied to this series.
  • #1
eljose79
1,518
1
Need help with euler,s transform of series...

I want to summ the series of general term:

(a(n)+1)Z**n from n 0 to infinite.. where a(n) is

a(n) 1 iff n is prime, and 0 elsewhere.

I have tried to use an euler,s transform to improve convergence..but is this trick legal for this series?..

In fact as you will have know the series is related to the sum of all primes for the function z**(x) where z would be a number...

i need to know if i can use euler,s tranform into my series or another trick to improve convergence.
 
Mathematics news on Phys.org
  • #2
What you wrote is
$$
S(z) = \sum_{n \in \mathbb{N}}Z^n + \sum_{p\text{ prime }}Z^p
$$
Looks as if it converges iff ##|Z|< 1##.
 

FAQ: Need help with euler,s transform of series

What is Euler's transform of series?

Euler's transform of series is a mathematical technique used to accelerate the convergence of a series. It involves taking the original series and transforming it into a new series that converges faster.

Why is Euler's transform of series useful?

Euler's transform of series is useful because it can help speed up the convergence of a series, making it easier to calculate the sum or approximate the value of the series. It can also help identify patterns or relationships between terms in the series.

How is Euler's transform of series calculated?

Euler's transform of series is calculated by taking the original series and multiplying it by a factor of n/(n+1), where n represents the position of the term in the series. This transformed series is then subtracted from the original series to create a new series. This process is repeated until the series converges.

What types of series can Euler's transform be applied to?

Euler's transform of series can be applied to both infinite and finite series. It is most commonly used for infinite series that converge slowly, but it can also be used for finite series to improve their accuracy.

Are there any limitations or drawbacks to using Euler's transform of series?

While Euler's transform of series can be a helpful tool, it may not always work for every series. In some cases, the transformed series may not converge or may converge to a different value than the original series. Additionally, the process of calculating the transform can be time-consuming and may not always result in a significant improvement in convergence speed.

Similar threads

I Extend Euler Product Convergence Over Primes: Basic Qs
Replies
1
Views
1K
I Infinite Series of Infinite Series
Replies
7
Views
2K
Insights Investigating Some Euler Sums
Replies
1
Views
2K
A Can You Prove the Series Is Periodic?
Replies
33
Views
2K
I Prove series identity (Alternating reciprocal factorial sum)
Replies
4
Views
2K
I Alternating Harmonic Numbers are cool, spread the word!
Replies
4
Views
1K
A An interesting series - what does it converge to?
Replies
5
Views
2K
A Memory issues in numerical integration of oscillatory function
Replies
3
Views
1K
I A thought about the Riemann hypothesis
Replies
7
Views
2K
MHB Euler/Riemann Point of Departure in Riemann's 1859 paper containing RH
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top