# 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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### When is a Fourier series analytic?

By Fourier theory, every continuously differentiable function $f : S^1 \to \mathbf C$ admits a unique, uniformly convergent Fourier expansion $$f(\theta) = \sum_{n\in \mathbf Z} a_n e^{in\theta}.$$ ...
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I have a simple problem: I need to evaluate the limit $x\rightarrow 1$ of the Jacobi Theta function 2 $$\Theta_2(m,x)=2x^{1/4}\sum_{k=0}^\infty x^{k(k+1)}\cos((2k+1)m)$$ when $m=0$, that to say $$\... 1answer 32 views ### Translations AND dilations of infinite series Sometimes, when working with infinite series, it's useful to add "dilated" or "translated" versions of the infinite series, term by term, back to the original. There are ways of making this rigorous ... 2answers 48 views ### When is a finite sum of powers of non-integer a rational number? [closed] Concretely, is there  b \in \mathbb R, n,k \in \mathbb N  such that  \sum_{i = n}^{n+k} b^i \in \mathbb Q ? 2answers 53 views ### Could the sum of powers of non-integers result in a whole number? [closed] Concretely, is there a  b \in \mathbb R  such that  \sum_{i \in I \subset \mathbb N} b^i \in \mathbb W ? 2answers 120 views ### radius of convergence \displaystyle \sum_{n=1}^{\infty}n! x^{n!} I just wondering radius of convergence following series$$ \sum_{n=1}^{\infty}n! x^{n!} \\ $$My 1st attempt is 'root test'$$ \sqrt[n!]{|a_{n!} |} =\sqrt[n!]{|n! |} =\sqrt[t]{t} \rightarrow 1 $$So, ... 1answer 10 views ### Power Series Solution for Differential Equation of Shifted Exponential Function I am trying to write a shifted exponential function as a power series. I am aware of the power series definition of the exponential function, i.e. e^{-x}=\sum_{n=0}^{\infty}\frac{(-x)^{n}}{n!}. ... 1answer 157 views ### Find the radius of convergence of the power series \displaystyle\sum_{n=0}^{\infty}a_nz^n, where a_{2k+1} = 2^k and a_{2k} = (1 + (1/k))^2 for k = 0, 1, 2, \dotsc I started off by doing the ratio test, but I know that the ration test is for |... 1answer 176 views ### What's the limit of coefficient ratio for a reciprocating power series? I have a question about the coefficient in the inverse of the power series. Assume$$ f=1-\sum_{i=1}^{\infty}(ck_i)x^i, $$where c and k_i are positive and 0<ck_i<1 for any i>0. ... 1answer 47 views ### radius of convergence of \sum_{n=1}^\infty \frac{2^nx^{n!}}{n} [closed] What is the radius of convergence of the following series?$$\sum_{n=1}^\infty \dfrac{2^nx^{n!}}{n}$$1answer 68 views ### Find the interval of convergence for these 3 power series I believe I need to use the root test and ratio test. I've solved the first two, but I'm not too sure I fully understand how to do these, so was hoping someone else could work them out so I could ... 2answers 114 views ### Identifying \sum\limits_{k=0}^\infty\frac{1}{k!}\frac{x^{k+6}}{k+6} [closed] I'm trying to prove this equality.$$\sum_{k=0}^\infty\frac{1}{k!}\frac{x^{k+6}}{(k+6)} {=} e^x(x^5-5x^4+20x^3-60x^2+120x-120)+120$$posted by: http://math.stackexchange.com/q/832368. How do I get ... 5answers 374 views ### Asymptotic behavior of \sum\limits_{k=1}^n \frac{2^k}{k} I'm looking for an asymptotic equivalent of$$\sum_{0 < k \le n} \frac{2^k}{k}$$as n \to \infty. A plausible candidate seems to be \frac{2^{n+1}}{n+1} (WolframAlpha plot, and the intuitive ... 1answer 126 views ### How to derive this interesting identity for \log(\sin(x)) [duplicate] I saw on SE that:$$\log(\sin x)=-\log(2)-\sum_{n=1}^{\infty}\frac{\cos(2nx)}{n} \phantom{a} (0<x<\pi)$$This is an extremely useful identity, as it helps solve:$$\int_{0}^{\pi} \log(\sin(x)...
I am working on the question below. It involves finding three different power series that meet certain conditions. (a) Find a power series $\sum_{n=0}^{\infty} a_nx^{n}$ that has a different ...
### Asymptotic expansion of $(1+\epsilon)^{s/\epsilon}$
I have taken the logarithm of this expression and computed the Taylor expansion of the $\log(1+\epsilon)$ term but by doing this we're required to calculate powers of this series when using the ...