# 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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### An easy question on complex

Let $\{u_{k}\}_{k=1}^{\infty}$ be a complex number sequence. If $\sum_{k=1}^{\infty}\lambda^{k}u_{k}=0$, for each $\lambda\in \mathbb{D}(0, 1/3)$(where the $\mathbb{D}(0, 1/3)~$denotes an open disc ...
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### Vanishing of Taylor series coefficient [duplicate]

I am solving previous year question paper some competitive exam. Give me some hint to solve the following problem. Let $f$ be an entire function. Suppose for each $a \in \mathbb{R}$ there exists at ...
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### If $\sum_{n=0}^{\infty}a_{n}x^n$ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n$ converges for $|x| < R$

I have the following statement: If $\sum_{n=0}^{\infty}a_{n}x^n$ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n$ converges for $|x| < R$ as well. I couldn't find a ...
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### Sum of sum of $k$th power of first $n$ natural numbers.

I was working on a problem which involves computation of $k$-th power of first $n$ natural numbers. Say $f(n) = 1^k+2^k+3^k+\cdots+n^k$ we can compute $f(n)$ by using Faulhaber's Triangle also by ...
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### Generalised Binomial Theorem Intuition

It was not until recently (why don't they teach it in secondary school?) that I've come across the Generalised Binomial Theorem, which from what I can tell is basically the same as the regular ...
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### Is this action of $\mathbb F[[x]]$ on $\bigoplus_{i=0}^{\infty}\mathbb F$ natural?

The title of my question has a field $\mathbb F$ in it, but to make sure I'm not losing anything, I would like to introduce my question in full generality. But still, I will be happy with an answer in ...