# Logarithmic Integral as Non-Convergent Series

## Theorem

The logarithmic integral can be defined in terms of a non-convergent series.

That is:

$\displaystyle \operatorname {li} \left({z}\right) = \sum_{i \mathop = 0}^{+\infty} \frac {i! \, z} {\ln^{i + 1} z} = \frac z {\ln z} \left({\sum_{i \mathop = 0}^{+\infty} \frac {i!} {\ln^i z} }\right)$

## Proof

From the definition of the logarithmic integral:

$\displaystyle \operatorname {li} \left({z}\right) = \int_0^z \frac {\mathrm d t}{\ln t}$

Using Integration by Parts:

 $\displaystyle \operatorname {li} \left({z}\right)$ $=$ $\displaystyle \left[{\frac t {\ln t} }\right]_0^z - \int_0^t t \, \mathrm d (\ln^{-1} t)$ $\displaystyle$ $=$ $\displaystyle \frac z {\ln z} + \int_0^z \frac {\mathrm d t} {\ln^2 t}$ $\dfrac 0 {\ln 0} = 0$ and Derivative of Function to Power of Function $\displaystyle$ $=$ $\displaystyle \frac z {\ln z} + \frac z {\ln^2 z} + \int_0^z \frac {2 \, \mathrm d t} {\ln^3 x}$ Integration by Parts, $\dfrac 0 {\ln^2 0} = 0$, $t \, \mathrm d \left({ln^{-2} t}\right) = -2 \,\ln^{-3} t$

This sequence can be continued indefinitely.

We will consider the nature of the terms outside and inside the integral, after a number $n$ of iterations of integration by parts.

Let $u_n$ be the term outside the integral.

Let $v_n$ be the term inside the integral.

After $n$ iterations of Integration by Parts as above, we have:

$\displaystyle \operatorname {li} \left({z}\right) = u_n + \int_0^z v_n \, \mathrm d t$
$\displaystyle u_0 = 0$
$\displaystyle v_0 = \frac 1 {\ln t}$

It follows that:

 $\displaystyle \operatorname {li} \left({z}\right)$ $=$ $\displaystyle u_n + \left[{t \, v_n}\right]_0^z - \int_0^z t \, \mathrm d \left({v_n}\right)$ $\displaystyle$ $=$ $\displaystyle u_n + \left[{t \, v_n}\right]_0^z + \int_0^z -t \, \mathrm d \left({v_n}\right)$

which gives us the recurrence relations:

 $\text {(1)}: \quad$ $\displaystyle u_{n + 1}$ $=$ $\displaystyle u_n + \left[{t \, v_n}\right]_0^z$ $\text {(2)}: \quad$ $\displaystyle v_{n + 1}$ $=$ $\displaystyle -t \cdot \frac {\mathrm d} {\mathrm d t} \left({v_n}\right)$
• By recurrence on $n$, with the following recurrence hypothesis:
 $\text {(\text{R.H.})}: \quad$ $\displaystyle v_n$ $=$ $\displaystyle \frac {n!} {\ln^{n + 1} t}$

When $n = 0$, we have:

$\displaystyle v_0 = \frac 1 {\ln t} = \frac {0!} {\ln^{0 + 1} t}$

which verifies the hypothesis.

By supposing true at $n$, we have at $n+1$:

 $\displaystyle v_{n+1}$ $=$ $\displaystyle -t \cdot \frac {\mathrm d} {\mathrm d t} \left({v_n}\right)$ from $(2)$ $\displaystyle$ $=$ $\displaystyle -t \cdot \frac {\mathrm d} {\mathrm d t} \left({\frac {n!} {\ln^{n + 1} t} }\right)$ from $(\text{R.H.})$ $\displaystyle$ $=$ $\displaystyle -t \cdot n! \cdot -\left({n + 1}\right) \cdot \frac 1 t \cdot \ln^{-\left({n + 1}\right) - 1} t$ Derivative of Function to Power of Function, Derivative of Natural Logarithm Function $\displaystyle$ $=$ $\displaystyle \frac {\left({n + 1}\right)!} {\ln^{n + 2} t}$ Definition of Factorial

So $(\text{R.H.})$ is verified at $n + 1$ if it is verified at $n$.

So it is proved for every $n \in \N$ (since it is true at $n=0$):

 $\text {(3)}: \quad$ $\displaystyle v_n$ $=$ $\displaystyle \frac {n!} {\ln^{n+1} t}$

By taking $(1)$, and inserting $(3)$ in, a new expression for $u_{n + 1}$ in function of $u_n$ (recursive expression):

 $\displaystyle u_{n + 1}$ $=$ $\displaystyle u_n + \left[{t \cdot \frac {n!} {\ln^{n + 1} t} }\right]_0^z$ $\displaystyle$ $=$ $\displaystyle u_n + \frac {z \, n!} {\ln^{n + 1} z} - \frac {0 \cdot n!} {\ln^{n + 1} 0}$ $\displaystyle$ $=$ $\displaystyle u_n + \frac {z \, n!} {\ln^{n + 1} z}$

That is, we can write by expanding:

 $\text {(4)}: \quad$ $\displaystyle u_{n + 1}$ $=$ $\displaystyle \sum_{i \mathop = 0}^n \frac {z \, i!} {\ln^{i + 1} z}$

$\blacksquare$