Help with infinite series sum{i=0=>inf} (x^i / (i)^2)

In summary, the person is seeking help for a closed form solution for a convergent infinite series involving the product of Poisson distributions. They receive assistance with the modified bessel function of the first kind, I_0\left(2 \sqrt{x}\right), and realize they should have been able to figure it out on their own.
  • #1
jacobcdf
5
0
Hi all,

I've been messing around with the product of Poisson distributions and was hoping someone could help me work out a closed form solution for the following convergent infinite series (given x > 0):

[tex]\sum^{\infty}_{i=0}[/tex] [tex]\frac{x^{i}}{i! \times i!}[/tex]​


Many thanks in advance,
Jacob.
 
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  • #2
[tex]I_0\left(2 \sqrt{x}\right)[/tex]

which is the modified bessel function of the first kind
 
  • #3
ice109 said:
[tex]I_0\left(2 \sqrt{x}\right)[/tex]

which is the modified bessel function of the first kind
Muchos gracias, amigo. That helps a lot. :)

Although now that I think about I should have been able to intuit that one on my own. I'm just getting old I guess. :(
 
Last edited:

1. What is an infinite series?

An infinite series is a mathematical expression that involves an infinite number of terms, where each term is added to the previous one. It is represented in the form of Σ (sum) followed by the general term of the series.

2. What is the general term of the infinite series sum{i=0=>inf} (x^i / (i)^2)?

The general term of this infinite series is x^i / (i)^2, where i represents the index or position of the term in the series. The value of x can be any real number, and the term (i)^2 in the denominator ensures that the series converges.

3. How do you find the sum of an infinite series?

The sum of an infinite series can be found using different techniques such as the geometric series formula, telescoping series, or power series. In this case, the series sum{i=0=>inf} (x^i / (i)^2) is a power series, so it can be solved by using the ratio test or the comparison test to determine its convergence.

4. What is the significance of the index in an infinite series?

The index in an infinite series represents the position of each term in the series. It starts from a specific value (usually 0 or 1) and increases by 1 for each term. The index helps to determine the value of each term and is also used in finding the sum of the series.

5. What is the purpose of using the ratio test in finding the convergence of an infinite series?

The ratio test is used to determine whether an infinite series converges or diverges. It involves taking the limit of the ratio of the (n+1)th term to the nth term as n approaches infinity. If the limit is less than 1, the series converges, and if it is greater than 1, the series diverges. In the case of sum{i=0=>inf} (x^i / (i)^2), the ratio test can be applied to determine the values of x for which the series converges.

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