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+--- Thread: Infinite products (/showthread.php?tid=267)
Infinite products - andydude - 04/06/2009
I just remembered a method I tried like 5 years ago, but haven't visited in a while.
Now that we have better power series expansions of tetration, for example, at \( x=(-1) \) we have
\(
\text{tet}(x) = \sum_{k=0}^{\infty}c_k(x+1)^k
\)
and using the definition, we can rewrite this as\text{tet}(x) = \sum_{k=0}^{\infty}c_k(x+1)^k
\)
\(
\begin{array}{rl}
\text{tet}(x)
& = \exp\left(\text{tet}(x-1)\right) \\
& = \exp\left(\sum_{k=0}^{\infty}c_kx^k\right) \\
& = \prod_{k=0}^{\infty} \exp(c_k{x^k}) \\
& = \prod_{k=0}^{\infty} a_k^{x^k} \\
\end{array}
\)
where \( a_k = e^{c_k} \)\begin{array}{rl}
\text{tet}(x)
& = \exp\left(\text{tet}(x-1)\right) \\
& = \exp\left(\sum_{k=0}^{\infty}c_kx^k\right) \\
& = \prod_{k=0}^{\infty} \exp(c_k{x^k}) \\
& = \prod_{k=0}^{\infty} a_k^{x^k} \\
\end{array}
\)
I wonder if this simplifies the coefficients or just makes things more complicated?
Andrew Robbins
RE: Infinite products - tommy1729 - 04/06/2009
andydude Wrote:I just remembered a method I tried like 5 years ago, but haven't visited in a while.
Now that we have better power series expansions of tetration, for example, at \( x=(-1) \) we have
\(and using the definition, we can rewrite this as
\text{tet}(x) = \sum_{k=0}^{\infty}c_k(x+1)^k
\)
\(where \( a_k = e^{c_k} \)
\begin{array}{rl}
\text{tet}(x)
& = \exp\left(\text{tet}(x-1)\right) \\
& = \exp\left(\sum_{k=0}^{\infty}c_kx^k\right) \\
& = \prod_{k=0}^{\infty} \exp(c_k{x^k}) \\
& = \prod_{k=0}^{\infty} a_k^{x^k} \\
\end{array}
\)
I wonder if this simplifies the coefficients or just makes things more complicated?
Andrew Robbins
that could have been my own post
i wondered about it too.
RE: Infinite products - bo198214 - 04/06/2009
andydude Wrote:\(where \( a_k = e^{c_k} \)
\begin{array}{rl}
\text{tet}(x) =
\dots & = \prod_{k=0}^{\infty} a_k^{x^k} \\
\end{array}
\)
I wonder if this simplifies the coefficients or just makes things more complicated?
To calculate the coefficients you either need on both sides a product or on both sides a sum. So I dont see how the product representation can be useful.