# 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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### Random walk - probability of first pass through zero.

In the proof that symmetric random walks end up regressing to the origin with probability $1$, I have found this didactic post on-line, where the power series of the probability mass function of the ...
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### How to use the generalized binomial theorem to produce the power series of $(1-x)^{1/2}$ [duplicate]

I am trying to see how to get from $\sqrt{1-x}$ to the power series $\displaystyle\sum_{m=0}^\infty\frac{-1}{2m-1}\,{2m \choose m}\,\frac{x^m}{4^m}$, ideally using the generalized binomial theorem. I ...
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### Asymptotic Expansions of a Generalized Hyper-Geometric Function

Let $t>0,x>0$, and $$\{a_1,a_2,a_3\}=\{2, 2, 9/8 - (i t)/2\}$$ $$\{b_1,b_2,b_3,b_4\}=\{1, 1, 3/2, 17/8 - (i t)/2\}$$ We are looking for the asymptotic expansions of a generalized hyper-...
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### Deciphering the theorem of perfect powers

Can someone help me understand what happened to this equation from the paper entitled Perfect Powers With All Equal digits but one...I don't understand the part when it lets a and c not equal to zero....
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### How to prove that $h''(x)$ has at most one zero on $(0,1)$.

$h(x)=1-\sum_{i=1}^{k-1}x^i+a_kx^k+\sum_{i=k+1}^\infty x^i$, where $|a_k|\le1$, is the power series of an analytic function. Prove that $h''(x)$ has at most one zero on $(0,1)$.
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### Convergence of a cosine series

Given that $$\sum_{n=1}^\infty a_n<\infty,$$ and that $$\lim_{n\to \infty}b_n=0$$ Is the series $$\sum_{n=0}^\infty a_nb_n^{-2}(1-cos(b_n))$$ necessarily convergent?
### Singularities of quotient of polynomials where the degree of the denominator $\ge$ the degree of the numerator $+2$.
Let the degrees of the polynomials $$P(z)=a_0+a_1 z+a_2 z^2+\cdots +a_n z^n \; (a_n \neq 0)$$ and $$Q(z)=b_0+b_1 z+b_2 z^2+\cdots +b_m z^m \; (b_m\neq 0)$$ be such that $m \ge n+2.$ Show that if ...