# Tagged Questions

Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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### $(x-x_0)^0$ in power series [duplicate]

When I first studied power series in high school, the teacher gave the following general definition: $$f(x)=\sum_{n=0}^{\infty}a_n (x-x_0)^n$$ He then proceeded to ...
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### Difficulty finding a power series representation

I have to find a power series representation and interval of convergence for $$f(x) = \frac{x-x^2}{(1+2x)^3}$$ Noting that $\frac{1}{1+2x}=\frac{1}{1-(-2x)}=\sum_{n=0}^\infty(-2x)^n$, I start taking ...
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### If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$.

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$. Now I know that $b_n=\dfrac{f^{(n)}(5)}{n!}$. I have tried various things but I think there is something wrong with my ...
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### Proving that a function is real-analytic

I try to solve the following exercise: Let $f:\mathbb{R}\to\mathbb{R}$ with $f(x):=\frac{1}{1+x^4}$. Prove that $f(x)$ is real analytic and compute the radius of convergence of it's Taylor series at ...
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### Complex power series expansion of $\frac{e^z}{1+z}$

I'm trying to find complex power series expansion of $\frac{e^z}{1+z}$ centered at $z=0$ and its radius of convergence. Here is my attempt: Since $e^z = \sum_{n=0}^\infty \frac{z^n}{n!}$, we can ...
### Maclaurin polynomial expansion of $y$ about 1?
Consider the differential equation $\frac{dy}{dx}=2x+\frac{y}{x}$, where $\frac{dy}{dx}=1$ when $x=1$. Find the first three non-zero terms in the Maclaurin polynomial expansion for $y$ about $x=1$....