# Absolute value of a Maclaurin series

I am trying to find out the absolute value of a Maclaurin power series of the below type: $$f(x) = a_0 + a_1 x+a_2 x^2+ \dots + a_n x^n$$, where $$x$$ is a complex number. I am interested to know the vakue of $$|f(x)| = |a_0 + a_1 x+a_2 x^2+ \dots + a_n x^n|$$.

I have tried looking around and found something relevant here. But, I am not sure if this relates to my problem and also what is the name of the equation $$|\lambda(z)| = 1 + Re(a_n z^n) + O(z^n)$$ and meaning of the term $$O(z^n)$$. Please let me know if there is any guidance in this regard. Thanks!

# Clarification

The Maclaurin series of a function $$f(z)$$ is a special case of the Taylor series...

$$T_f^{z_0}(z) = \sum_{k=0}^{\infty}{\frac{f^{(k)}(z_0)}{n!}(z - z_0)^k}$$

Where $$z_0 = 0$$...

$$T_f^{0}(z) = \sum_{k=0}^{\infty}{\frac{f^{(k)}(0)}{k!}z^k}$$

It is important to make the correction that $$f(z)$$ does not equal, it approximates the finite sequence...

$$f(z) \approx a_0 + a_1z + a_2z^2 + \ldots + a_nz^n$$

It is equal to the infinite sum...

$$f(z) = a_0 + a_1z + a_2z^2 + \ldots + a_nz^n + \ldots$$

So, the absolute value of $$f(z)$$ would be...

$$\left|T_f^{0}(z)\right| = \left|\sum_{k=0}^{\infty}{\frac{f^{(k)}(0)}{k!}z^k}\right|$$

# Example

What is the Maclaurin series for $$g(z) = e^z$$?

$$T_g^{0}(z) = \sum_{k=0}^{\infty}{\frac{g^{(k)}(0)}{k!}z^k} \\\implies e^z = \sum_{k=0}^{\infty}{\frac{z^k}{k!}}$$

What is the Maclaurin series for $$f(z) = \left|g(z)\right|$$?

$$T_f^{0}(z) = \left|\sum_{k=0}^{\infty}{\frac{g^{(k)}(0)}{k!}z^k}\right| \\\implies\left|e^z\right| = \left|e^{\operatorname{Re}(z) + i\operatorname{Im}(z)}\right| = \left|e^{\operatorname{Re}(z)}\right|\left|e^{i\operatorname{Im}(z)}\right| = e^{\operatorname{Re}(z)} \\\implies\left|e^z\right| = \sum_{k=0}^{\infty}{\frac{\operatorname{Re}(z)^k}{k!}}$$

• Sorry to ask naive questions. But can you please let me know why $|e^{iIm(z)}|$ = 1? Commented Nov 9, 2019 at 21:17
• @NIT_GUP The answer is made clear by using Euler's formula (math.stackexchange.com/q/721784/490122) Commented Nov 9, 2019 at 21:28