# Questions tagged [power-series]

Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n (x-c)^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex.

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### Proving $f(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + …$ is continuous for a fixed $x_0 \in (-1,1)$ by using the Weierstrass M-test

I am trying to prove that $f(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + ...$ is continuous for a fixed $x_0 \in (-1,1)$ by using the Weierstrass M-test. Now, the solution in the book is ...
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### Is the radius of convergence related to the ratio limit or half of the interval of convergence?

I have a series $S$ with general terms $a_n=\frac{(-1)^n(x-1)^n}{(2n-1)2^n}$, $n\ge 1$: $$S = \sum_{n=1}^\infty \frac{(-1)^n(x-1)^n}{(2n-1)2^n}$$ Finding the ratio $\left|\frac{a_{n+1}}{a_n}\right|$ ...
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### Polynomial bound for complex exponential function

Can you help me prove that $\vert \exp(it) - 1 -it \vert \le t^2$, where $t > 0$? Thank you in advance.
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### Calculate a meromorphic function on $\mathbb{C}$ s.t. $g(z) = \sum_{n\geq0}(n+1)T^n$

I have the following power series $f(T)=\sum_{n\geq0}(n+1)T^n$ and I have to calculate a meromorphic function $g$ on $\mathbb{C}$ such that $\forall z \in B_1(0), g(z)=f(z)$ I have tryed the following:...
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### Root test for complex series and cancelling powers with absolute values

The root test for convergence of a complex power series is given as $$\lim_{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|} = L$$ If $a_n = \frac1{(1+i)^n}$ then I read that when applying the root ...
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### Find multiplicative inverse of formal power series

I have the following power series: $f(T) = \sum_{n\geq 0}(n+1)T^n \in \mathbb{C} [T]$ and I have to compute the multiplicate inverse. I've tryed to use the formula to calculate the coefficients of the ...
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### Finding the power series solution to ${(x^2+1)y'' + xy' + 2xy = 0}$ [closed]

Find the power series solution to the following initial value problem about the point ${x=0}$: $${(x^2+1)y'' + xy' + 2xy = 0}$$ subject to $${y(0) = 2}$$ $${y'(0) = 3}$$ My attempt: I tried to find ...
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### Show that $\sum_{n = 2}^{\infty} \left(\frac{(-1)^{n}}{(n-1)n} - \frac{9(-1)^n}{2\cdot 3^n}\right)x^n > 0$ for positive $x$.

How can one prove that $$\sum_{n = 2}^{\infty} \left(\frac{(-1)^{n}}{(n-1)n} - \frac{9(-1)^n}{2\cdot 3^n}\right)x^n > 0\,?$$ this inequality resulted after expanding the difference of two functions ...
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### $\sum_{n=1}^{\infty}(a_{n}-a_{n-1})x^n$ and $\sum_{n=0}^{\infty}a_nx^n$ has the same convergence radius [closed]

Proof: Suppose power series $\sum_{n=0}^{\infty}a_n x^n$, if series ${a_n}$ is divergent. Then $\sum_{n=1}^{\infty}(a_{n}-a_{n-1})x^n$ and $\sum_{n=0}^{\infty}a_nx^n$ has the same convergence radius. ...
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### Power series problems

How can I find the power series expansion of following functions. I don't have any idea, these seem intimidating. Please help how to proceed. $(x+\sqrt{1+x^2})^a$ $\sqrt{\frac{1-\sqrt{1-x}}{x}}$
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### Power Series ODE Question - Final Step

I just started learning how to use the power series method to solve ordinary differential equations and this is one of the first questions we were asked. I've managed to get it right up to the last ...
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### Behavior of the wavefunction at points where potential is singular

I am a physics student trying to understand the behavior of a wavefunction $\psi(x)$ at points where the potential $V(x)$ is singular. In Griffith's Inroductuion to Quantum Mechanics book, he wrote ...
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### Can I find a power series expansion of this function without a Taylor series?

I was asked to find the power series expansion of $f(x) = \frac{x}{\sqrt{x^2+4}}$ about $x = 0$. Is there a way to do a power expansion without finding the Taylor series? Deriving this function ...
I was asked to multiply these two power series expansions: ie. $p(x)q(x)$ $$p(x) = 2(x − 4) − 7(x − 4)^2 + 5(x − 4)^3 + . . .$$ $$q(x) = 4 − 3(x − 4) + (x − 4)^3 + . . .$$ and I was wondering if there'...