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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7
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5answers
507 views

Why the radius of convergence and not “areas of convergence” for power series?

My calculus is quite rusty and I'm trying to rebuild it on an intuitive basis. Currently, I am looking at power series and have trouble understanding the radius of convergence. I am comfortable with ...
1
vote
2answers
46 views

approximate $\int_{0}^{0.5}{\frac{\sin(x)}{x}}dx$

By using Maclaurin series, approximate the value of $$\int_{0}^{0.5}{\frac{\sin(x)}{x}}dx$$ to within an error $0.0001$, where $x$ is in radians. My attempt: Since we know the Maclaurin series of ...
0
votes
1answer
47 views

Radius of convergence and sum of alternating series $1 - z + z^2 - z^3 + \ldots $

I have a (complex) function represented by the power series \begin{equation*} L(z) = z -\frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} \ldots \end{equation*} which I have tried to represent (perhaps ...
1
vote
3answers
70 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 $\tan x$ ...
0
votes
0answers
25 views

power series steps help

Can someone show me how to do this problem? I really need a walk through of the steps if possible. Find a power series representation for the function and determine its radius of convergence. $$ ...
3
votes
1answer
68 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
vote
1answer
22 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
votes
1answer
20 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
76 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?
1
vote
1answer
56 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
vote
1answer
40 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 ...
1
vote
2answers
50 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
votes
1answer
23 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 ...
0
votes
3answers
26 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 ...
0
votes
0answers
22 views

Help with simplification rules form sums and integrals.

IF you had a power series with summation notation and an integral what expressions would you be able to pull outside the integral and which would you be able to pull outside the sum.
0
votes
1answer
20 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.
-1
votes
2answers
75 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
0
votes
0answers
14 views

Recursive relationship for Peano Baker Series

The Peano Baker Series is a integral has the following form $$\varPhi(h,0)=I+\intop_0^h G(t_{1}) \, dt_1 + \intop_0^h G(t_1) \intop_0^{t_{1}} G(t_2) \, dt_2 \, dt_1 + \intop_0^h G(t_1) ...
1
vote
0answers
69 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 ...
-1
votes
2answers
42 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
1
vote
3answers
40 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 $x=2$.
0
votes
4answers
30 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 $x^n$
0
votes
2answers
22 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
votes
1answer
42 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 ...
1
vote
1answer
24 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
votes
2answers
75 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 ...
0
votes
0answers
48 views

how would i simplify this into an identity?

$$ B_{n,k}^{f\ln(g)} = B_{n,k}\left(\frac{d}{dx}[f(x)\ln(g(x))], \frac{d^2}{dx^2}[f(x) \ln(g(x)), \cdots, \frac{d^{n-k+1}}{dx^{n-k+1}}[f(x) \ln(g(x))]\right) $$ We know that: $$ B_{n,k}^{f\ln(g)} = ...
0
votes
1answer
14 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)!} + ...
1
vote
2answers
52 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
vote
1answer
46 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
vote
2answers
44 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
votes
1answer
49 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 ...
0
votes
0answers
17 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
vote
3answers
49 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 ...
0
votes
1answer
11 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 = ...
1
vote
2answers
46 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?
0
votes
0answers
21 views

How do i take the inverse cauchy product of the following summation?

Well i start out by defining the following partial bell polynomial as: $$ B_{n,k}^u = \frac{n!}{k!} [t^n] \left(\sum_{m \geq 1} \frac{d^m}{dx^m}[u] \frac{t^m}{m!} \right)^k $$ Where $[t^n]$ is the ...
8
votes
1answer
145 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.
0
votes
0answers
39 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 ...
-4
votes
1answer
76 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
vote
1answer
75 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 ...
2
votes
1answer
45 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
83 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
47 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
69 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 ...
1
vote
1answer
114 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
votes
1answer
50 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
votes
2answers
26 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 ...
4
votes
3answers
106 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 ...
5
votes
1answer
158 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 ...