# 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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### Find an analytic continuation

Let $f(z)=\sum_{j=0}^{\infty}z^j$ for $|z|<1$. For what values of $\alpha$ ($|\alpha|<1$) does the Taylor expansion of f(z) about $z=\alpha$ yield a direct analytic continuaton of f(z) to a disk ...
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### If the Power Series converges at x, which must be true?

I'm currently reviewing for tomorrow's Calculus BC exam but I got stuck on this one problem. $\sum_{n=0}^{\infty} a_n (x-3)^n$ converges at $x = 5$. Which of the following must be true? a. ...
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### Series expansion for monotone, bounded functions

I would like to know if there is a series expansion for monotone and bounded functions, where all functions of the orthonormal basis are monotone as well. I.e. suppose that we have a function $f(x)$ ...
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### Solving applications of power series.

The region bounded by the curves $$y = \frac{\sin(x^2)}{x}$$ $$y = 0, x = \frac{1}{2}$$ is rotated around the y-axis. Find the volume of the solid of revolution with accuracy to within $10^{-2}.$ ...
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### Finding an analytic function

enter image description here I cannot find any such function. Also, why would a function that is analytic at 0 following these criteria not be analytic on (-2,0). Thanks in advance for your help.
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### Integral analog of geometric series

We all know that $$\frac{1}{1-z}=\sum_{m=0}^\infty z^m\ ,$$ for $|z|<1$. The challenge I would like to pose is: find (possibly as simple and elegant) integral representations (as many as you can) ...
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### Did Feynman mentally compute $\sqrt[3]{1729.03}$ by linear approximation?

In the biopic infinity'' about Feynman. (11:48~15:50) Feynman compute $\sqrt[3]{1729.03}$ by a mental calculation. I guess that he use the linear approximation. That is, he observe that $1728=12^3$....
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### The Method of Frobenius

I'm learning about the method of Frobenius and solutions about singular points. But the class didn't cover the $2^{nd}$ and the $3^{rd}$ case below. Could someone please explain the parts that I ...
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### Evaluation of r^3 from zero to N (Sigma Notation)

I have to evaluate the following expression, $\sum_{n = 0}^{9} (n^3 -1)$. I know that $\sum_{n = 1}^{9}(n^3 -1)$ is given by $\frac{1}{4}(9)^2(9+1)^2 - 9$ But, how do I do this from zero to ...
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### Show that any polynomial of odd degree 2n+1: $f(x)=\sum_{k=0}^{2n+1} a_kx^k$, $a_{2n+1}\neq0$ has at least one real root.

Show that any polynomial of odd degree 2n+1: $$f(x)=\sum_{k=0}^{2n+1} a_kx^k$$ $a_{2n+1}\neq0$ has at least one real root. I would like to prove this using IVT, how would I go about starting ...
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### For what values of $k$ to both of the following series converge?

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
Please help me understand this: Solve $(x+1)y''-(2-x)y'+y =0$ First, since $x_0=0$ is an ordinary point, it can be guaranteed that we can find two independent power series solutions centered at $x_0=... 1answer 48 views ### Radius of Convergence of$\sum_{n = 0}^{\infty} \frac{(-1)^nn!x^n}{n^n}$I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ... 1answer 26 views ### How would I show$|x| \le 1$given the equation for$x$the expression in the equation? The expression is$x = \sin(\theta /2)$. I am asking how would I show that$\sin(\theta/2)\le1$based on the expression? I already know that the biggest$\sin$will ever get is$[-1, 1]$which is the ... 3answers 128 views ### How is the last “=” true? How can the last equality be true? $$G(u)=\frac{g}{(1+u)^g-1}-\frac1u=\frac{g}{gu + \cdots + u^g} - \frac{1}{u}=\sum_{k=0}^\infty c_ku^k$$ 1answer 35 views ### What is the Laurent expansion of$f(z)=\frac1{z-3}$? What is the Laurent expansion of$f(z)=\dfrac1{z-3}$? In the region,$|z-3|>0$? I just computed the Laurent expansion in the region$|z|>3$by dividing the denominator by$\dfrac1z$and ... 5answers 56 views ### Does this power series$\sum_{n=0}^{\infty} \frac{n^n}{(n!)^2}x^n$converge for all$x$? Does this power series$\sum_{n=0}^{\infty} \frac{n^n}{(n!)^2}x^n$converge for all$x$? It was told to me that the series does converge for all$x$, however I have investigated with a computer ... 0answers 22 views ### Second Order Linear Non-Homogeneous DE solution with Power Series$x^2y'' - 4xy' + 6y = x^2 \cos x$My instructor wants me to solve the above equation using power series and another method, and then to confirm the results are the same This equation does not have constant coefficients and a can't ... 1answer 43 views ### Help with generating functions I've got two questions. I'm trying to extract the "coefficients" of a power series. I think my terminology is incorrect here but here is what I mean. Here are some examples A(Z) = 1/(1-Z) &... 2answers 44 views ### Solve$y'=e^{x^2}y$(with$3$terms only) in using power series Solve$y'=e^{x^2}y$(with$3$terms only) in using power series. I know that$e^{x^2}=\sum_{n \geq0} \frac{x^{2n}}{n!}$, but I don't know how to find the coefficients$a_n$in considering$y=\sum_{n ...
Having the equation $$x^{2}y''+xy'+x^{2}y=0$$ I get the indicial equation at get r=0, and am left with the equation. r^{2}a_{0}x^{r}+(r^{2}+2r+1)a_{1}x^{r+1}+\sum^{\infty}_{0}\big[[(n+r+2)(n+r+1)+(...