# 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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### What is the radius of convergence of $\sum_{k = 0}^{\infty} 3^{k^2}x^{k^2}$

Where I'm at right now: $a_k = 3^{k^2}$, so we do $\frac{1}{\lim sup_{k\to\infty}(3^{k^2})^{\frac{1}{k}}}$, which is $\frac{1}{3^k}$, but that doesn't seem right. I suppose from there, I go through ...
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### 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 ...
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### Calculate the radius of convergence of $\sum^\infty_{k=1} \frac{(2k-1)^{2k-1}}{2^{2k}(2k)!}x^k$

I need some assistance on calculating the radius of convergence: $\sum^\infty_{k=1} \frac{(2k-1)^{2k-1}}{2^{2k}(2k)!}x^k$ I tried the quotient criteria: ...
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### Complex power series : Radius of Convergence

Could anyone please suggest me how to deal with these questions (Complex Variable) : Note that all problems are in $\mathbb{C}$. 3.1 Determine the radius of convergence $\rho$ of each of the ...
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### Series approximation to $\int_0^1\sqrt{\frac{2x+3}{2u^3-(2x+3)u^2+2x+1}}du$

I have figured out by graphing that, for small $x$: $$\int_0^1\sqrt{\frac{2x+3}{2u^3-(2x+3)u^2+2x+1}}du\approx\log(1/x)+\pi/2+O(x)$$ However, I am unable to prove that this is the case. As $x\to 0$ ...
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### Is there a formula for $\sum_{n=0}^{+\infty} q^{n^3}$?

When I studyied the representation of integers as sum of squares, I found that the most powerful tool is the Jacobi Triple Product, in fact this amazing identity allows us to find more useful ...
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### How to find the Number of factors, if sum of the factors are given?

A number is expressed in terms of $(2^m\times3^n)$, Find the value of $(m,n)$ if sum of all factors of a number is $124$.
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### Proving the sum of squares of sine and cosine using the Cauchy product formula

Here are the power series of sine and cosine: $$\sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac {x^{2n+1}} {(2n+1)!}$$ and $$\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac {x^{2n}} {(2n)!}$$ How can it be ...
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### $e^{x} > 1$ and $0 < e^{x} < 1$

So $$\exp(x) := \sum_{n=0}^{\infty} \frac {x^n} {n!}$$ How to prove that $\exp(x) > 1$ when $x > 0$ and moreover $\exp(x) < 1$ when $x<0$ Is it possible with induction? Or must I use ...
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### What can be said about the limit of a converging infinite polynomial?

Suppose we have the following polynomial of infinite order: $f(x) = a_0+a_1x+a_2x^2+...=\sum_{n=0}^{\infty}a_nx^n$ Also suppose that $f(x)$ converges to some limit $f^*$ as $x\nearrow\infty$, i.e. ...
### Taylor series expansion of $e^{x+y}$ about the point $(0,1)$
My question is: what is the Taylor series expansion of $e^{x+y}$ about the point $(0,1)$? I think the standard $e^{x+y} = 1 + x+y + 1/2(x+y)^2$ ... doesn't apply here. Thanks in advance