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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3answers
908 views

Working out $\tan x$ using sin and cos expansion

Using only the series expansions $\sin x = x- \dfrac{x^3} {3!} + \dfrac{x^5} {5!} + ...$ and $\cos x = 1 - \dfrac{x^2} {2!} + \dfrac{x^4}{4!} + ...$ Find the series expansions of the $\tan x$ ...
3
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1answer
82 views

Is my proof that $\frac{\pi}{4}=\sum\limits_{n\geq 0}(-1)^n \frac{1}{2n+1}$ correct?

Respected All I was trying to prove that $$\sum_{n\geq 0}(-1)^{n} \frac{1}{2n+1}=\frac{\pi}{4}$$ What I tried to show like this. We know $$\frac{1}{1+x^2}=(1+x^2)^{-1}=\sum_{n\geq 0}(-1)^nx^{2n}, |...
1
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1answer
32 views

Power series confusion

I'm having trouble with power series. Can someone walk me through this? My biggest problem is always figuring out what I need to break apart. Find a power series representation for each function ...
0
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1answer
43 views

Difference between a convergent series and an asymptotic series?

Can someone let me know the difference between a convergent series and an asymptotic series with an example? Can both the series be the same at some situations? In what situations an asymptotic series ...
5
votes
1answer
271 views

Debye Function Integral (BlackBody)

Show that $$ \int^{\infty}_{0} \frac{x^{3} \, dx}{e^{x}-1} = \frac{\pi^{4}}{15} $$ by expanding the integrand in powers of $e^{-x} $ and integrating term by term. Could anyone help with this one?
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1answer
60 views

Infinite series $\sum_{n=1}^{\infty}nx^{n+1}$ does not comply to any of my (known) tests

I am attempting to find the interval of convergence for $$\sum_{n=1}^{\infty}nx^{n+1}$$ The lower bound, x = -1, would be tested by determining if $$\sum_{n=1}^{\infty}n(-1)^{n+1}$$ diverges. ...
1
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1answer
93 views

how to understand Taylor's inequality intuitively?

I am learning the Taylor Series at the moment and I am trying to figure out how to understand Taylor's inequality intuitively. I know you can integrate repeatedly and prove the inequality is correct....
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2answers
51 views

Find the function that equals to $1-x^3+x^6-x^9+ \cdots$

Find the function that equals to $1-x^3+x^6-x^9+ \cdots$ for all $|x| < 1$ I know that $\frac{1}{1+x} = 1-x+x^2-x^3+...$ But I couldn't find the pattern here
0
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1answer
37 views

Power series expansion of f(x)=1/(1-x) around x=0 and x=-1

For the power series expansion of the function $f(x)$ I worked out the at $x=0$ the power series expansion is $$1(x-0)^n$$ and at $x=-1$ the power series expansion is $$\left(\frac{1}{2^n}+1\right)\...
0
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3answers
33 views

Power series expansion using Taylors Theorem.

So the function $f(x)=3x^2-6x+5$ needs to be written as a power series expansion around $x=a$ and the goal is to show $x=a$ is $f(x)$ for every $a$. So I started off by finding up to the third ...
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1answer
22 views

Help with general power series concept

If f(x) is some general polynomial, what will the power series expansion of f(x) be. Is there a set rule for finding the power series of polynomials.
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2answers
80 views

How can I solve the following differential equation without use power series [closed]

Let we have the following differential equation $$y''-xy'=e^{-x}$$ how can I solve this differential equation without use power series
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0answers
70 views

How to solve $1 = \sum_{p \text{ prime}} x^{-p-1}$?

As the title says, I am trying to solve the equation $$1 = \sum_{p \text{ prime}} x^{-p-1}$$ and I'm not really sure where to begin. I got this from an exercise in a book and apparently there is a ...
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2answers
47 views

How can I solve the following differential equation [closed]

How can I solve the following differential equation by power series near the point $z=1$ $$(z^2-2z+2)w''+2(z-1)w'=0$$ Then I have to find the radius of convergence of the solution
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3answers
53 views

How to solve power series expansions.

The function is $f(x)=1/(1-x)$ and it asks to find a power series expansion expanded around $x=a$, which would be the general expansion as well as around $x=0$ and $x=2$.
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4answers
36 views

Power Series representation of $\frac{x^3}{(3x+4)^2}$

How do you do this? I have an exam in 2 hours and I know this type will be on it and I have no clue. We were taught to base it off the power series of $x^n$
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2answers
28 views

Estimate on rate of growth of a power series

Given two sequences $(a_k),(b_k)$ with $a_k\geq0,b_k>0$ such that the power series $\sum_{k=0}^\infty a_k b_kr^{k}$ and $\sum_{k=0}^\infty a_kr^k$ converge for each $r>0$. My question now is: ...
0
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1answer
73 views

An entire function is a polynomial iff the Taylor expansion around $0$ converges uniformly

Let $g:\mathbb{C} \to \mathbb{C}$ an entire function. Prove that the Taylor expansion around $0$ converges uniformly in all $\mathbb{C}$ if and only if $g$ is a polynomial. 1/2 PROOF I think I have ...
1
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1answer
72 views

Multiplying and factoring in Formal Power Series

I'm working with some formal power series in my homework. Somewhere in the middle of my hw problem I reach a point where I would really like to factor, but I'm not sure if I can. Suppose $F_k$ ...
2
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2answers
79 views

Prove $f(x)=g(x)$ for all $x \in\mathbb{R}$

If $$f(x)=\sum_{n=0}^\infty\frac{x^n}{n!}, x\in\mathbb{R}$$ and $$ g(x) = 1 + \int_0^x f(t) \,dt $$ prove that $g(x)=f(x)$ for all $x\in\mathbb{R}$ and prove that $f$ is differentiable on $\mathbb{R}...
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1answer
15 views

Verifying whether an expression equals $\frac{1}{x}$

The derivative of some expression turned out to be: $$\frac{e^x}{x}(1 -\frac{1}{2x}) + e^x\sum_{n=2}^{\infty}\frac{x^{n -1}}{n!} + \frac12 \sum_{n =3}^{\infty}\frac{x^{n -2}}{n(n-2)!} + e^{-x}\...
1
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2answers
58 views

Solving $y'(x)-2xy(x)=2x$ by using power series

I have a first order differential equation: $y'(x)-2xy(x)=2x$ I want to construct a function that satisfies this equation by using power series. General approach: $y(x)=\sum_0^\infty a_nx^n$ ...
1
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1answer
59 views

How do I extrapolate summation notation from a given series?

I am currently working on the power series for a homework assignment. I have to find the radius of convergence for the function $$\frac{10}{1+64x^2}$$ By setting up the $$\frac{1}{1+64x^2}$$ part ...
1
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2answers
46 views

Finding the co-efficients of this power series

I am required to find the co-efficients of this power series: $2x\ln(1+2x)$ I approached the problem by considering the $\ln(1+2x)$ part as the integral of $2/(1+2x)$ and applied the geometric series ...
3
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1answer
271 views

Bounds on Maclaurin series of $e^{-x^2}$

This is a problem from a textbook: By taking the 4th degree Maclaurin polynomial for $e^{-x^2}$ find an approximation to $\int^1_0 e^{-x^2} \text{dx}$. Place bounds on the error in this ...
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0answers
22 views

Maclaurin polynomial error term

this is a problem from a textbook, What degree Maclaurin polynomial of $e^x$ must be taken to guarantee an estimate of $e$ to within $1 \times 10^{-6}$ The answer from textbook is $n=17$, but I ...
1
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3answers
51 views

Has this infinite sum $\sum _{i=1}^{\infty } p^i \log (b i+a)$ any known solution?

I am wondering if exist a known solution for this kind of infinite sum $$ \sum _{i=1}^{\infty } p^i \log (a i + b) $$ where $p,a,b$ are real and $p\leq 1$. ...or even an approximation of the exact ...
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1answer
20 views

Derivative of Bessel J series… Do I reindex my summation?

Okay, short question: what happens to my index upon differentiation and why? This is a small step in a larger proof I'm working on... Given the series representation of Bessel J $$J_n = \sum_{k=0}^{...
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2answers
53 views

Question on power, If 2x^2x^2x^2x… =4 Solve for x

I've seen this random example, in which can anyone give me clue how to solve for $ x $ here?
8
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1answer
148 views

Expressing ${}_2F_1(a, b; c; z)^2$ as a single series

Is there a way to express $${}_2F_1\bigg(\frac{1}{12}, \frac{5}{12}; \frac{1}{2}; z\bigg)^2$$ as a single series a la Clausen? Note that Clausen's identity is not applicable here.
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0answers
55 views

How expand an equation in powers of two variables?

Let $$ \varphi=\int\frac{dr}{r^2\sqrt{\frac{1}{b^{2}}-\left(1-\frac{s}{r}\right)\frac{1}{r^{2}}}} $$ Is it possible to expand the above equation in powers of $\frac{s}{r}$?. I know that after ...
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1answer
81 views

Find the value of Taylor coefficient $a_5$ for the function $ \int_0^x (e^{-t^2}+\cos t) \, dt$

Find the value for $a_5$ If $ \int_0^x (e^{-t^2}+\cos t) \, dt$ has the power series expansion $\sum_1^\infty a_nx^n$, then find $a_5$ up to three correct decimal places. I think it is a Taylor ...
1
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1answer
166 views

Find the Laurent expansion for $f(z)=\frac{\exp{1/z^2}}{z-1}$ about $z=0$.

Find the Laurent expansion for $f(z)=\frac{\exp{(1/z^2)}}{z-1}$ about $z=0$. I was able to determine the series for each of the factors. We have $$e^{1/z^2}=1+\frac{1}{z^2}+\frac{1}{2!z^4}+\frac{1}{...
2
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1answer
130 views

Maclaurin Series: Complex Analysis

Question: Use the representation $\sin z = \sum\limits_{n=0}^\infty (-1)^n\frac{z^{2n+1}}{(2n+1)!}$, $|z|<\infty$ to write the Maclaurin series for the function $f(z) = \sin z^2$ and point out how ...
4
votes
1answer
149 views

Find the Laurent Expansion of $f(z)$

Find the Laurent Expansion for $$f(z)=\frac{1}{z^4+z^2}$$ about $z=0$. I have found the partial fraction decomposition $$f(z)=\frac{1}{z^4+z^2}=\frac{1}{z^2}-\frac{1}{2i(z-i)}+\frac{1}{2i(z+i)}.$$ ...
0
votes
2answers
95 views

Proof for multiplication of two power series

Prove that $(\sum_{k=0}^\infty u^k)^2=\sum_{k=0}^\infty (k+1)u^k$ when |u|<1. This is a proof I need for a larger proof I was doing. I am stuck on this, so I was not able to make any notable ...
3
votes
1answer
74 views

Irreducibility in $k((t))[y]$

Let $k$ be an algebraically closed field of char $0$ and suppose $f(y) \in k[y]$ (need not be monic). Let $t$ be an indeterminate and consider the fraction field $k((t))$ of power series ring $k[[t]]$....
1
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1answer
121 views

Real Analysis II — power series

If f(x)= $\int_0^x \frac {ln(1+u)}{u}$ du, find a series for f(x) and calculate the approximate value of f(0.1). Use the error upper bound for approximating an alternating series to give an upper ...
0
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1answer
65 views

Determine convergence of power series of triple integral

Does this power series converge? I think it does, but how to prove it? $$ \sum_{i=0}^\infty \int \int \int 3^{-i}\left(\cos(\pi x) + \cos(\pi y) + \cos(\pi z) \right)^idxdydz, $$ where the ...
0
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2answers
40 views

Power series representation of xln(3x+5)

I get to the point $\sum_{n=0}^{\infty }\frac{(-1)^n3^{n+1}x^{n+2}}{(n+1)5^{n+1}}$ by using the geometric series and integrating etc. But I looked up the answer and it is what I have plus the term xln(...
4
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3answers
145 views

Proving positivity of the exponential function

Question. Without using the semigroup property ($\mathrm{e}^{x}\mathrm{e}^{y}=\mathrm{e}^{x+y}$), how can we show that $\mathrm{e}^{x}>0$ for all $x\in\mathbb{R}$ only by using the series expansion?...
5
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1answer
199 views

Power series expansion of Blaschke product

Suppose $B$ is a Blaschke product with at least one zero off the origin, and $B(z)=\sum_{k=0}^\infty {c_kz^k}$. Is it possible that $c_k\ge0$ for all $k=0,1,\ldots$? My try: Since $B(z)$ takes real ...
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1answer
31 views

power series where $f(z)=e^z$ and $z_0=1$

How do i find the power series of the form: $$\sum_{n=0}^{\infty}a_n ({z-z_0)}$$ where $f(z)=e^z$ and $z_0=1$ using the geomatric series currently i have that it equals $$\sum_{n=0}^{\infty} ({e^z-...
2
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2answers
59 views

What method was used here to expand $\ln(z)$?

On Wikipedia's entry for bilinear transform, there is this formula: \begin{align} s &= \frac{1}{T} \ln(z) \\[6pt] &= \frac{2}{T} \left[\frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \...
0
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1answer
308 views

Given a power series with interval of convergence $(-1,1]$, construct a series with another given interval of convergence

Suppose that you have a power series $$\sum_{n=1}^\infty (a_nx^n)$$ whose interval of convergence is $(-1,1]$. A) Using the same numbers $(a_n)$, come up with a new power series whose interval of ...
0
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0answers
50 views

Why does the limit of the p-series converge to a non-zero constant?

Here, the limit of the integral apparently converges to infinity as x goes to infinity. What I don't understand though, is how could it possibly converge to a constant of 5/7? If you multiply 5/7 by ...
1
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1answer
44 views

Series solution

Given the differential equation $2(1-x)y''-3y'+\frac{y}{x}=0$ and in standard form: $y''-\frac{3}{2(1-x)}y'+\frac{1}{2x(1-x)}y=0$ I want to find the series solution for the larger root $σ = 1$ of ...
1
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2answers
953 views

Finding a recurrence relation, first few terms of power series solution to differential equation

I'm attempting to find a recurrence relation and the first few terms of a power series solution for the differential equation: $$(1-x^2)y \prime\prime - 2xy\prime + \lambda y = 0$$ Where $\lambda$ ...
2
votes
0answers
50 views

Finding a Taylor Expansion for the following:

I have: $$\frac{1}{1-z}$$ for $z_0=i$. I have no idea how to do the Taylor Series expansion of this, around $z_0=i$, and then show it summation form. I have: $\frac{1}{1-z} = \frac{1}{1-i}+\frac{(...
1
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1answer
278 views

General solution of $(1-x^2)y''-2y=0$ about $x_0=0$?

I've expanded this differential equation as a series to obtain the recurrence relation $$a_{n+2}=\frac{a_n(n^2-n+2)}{n^2+3n+2}.$$ I don't know how to find $a_n$ in terms of $a_0$ and $a_1$ so that I ...