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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5
votes
2answers
474 views

Equality with Euler–Mascheroni constant

While trying to prove integral with exponential function and logarithm in an alternative way, I came to this solution: $$\sum_{k=0}^{+\infty}(-1)^{k+1}\frac{\log (k+1)+\gamma }{(k+1)}.$$ As both ...
4
votes
1answer
86 views

Short form of few series

Is there a short form for summation of following series? $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ ...
4
votes
1answer
154 views

Estimate the scale of $e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m}$

I would like to estimate the scale of the following series, $$S(m,t)=e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m},$$ where $e$ is the base of ...
4
votes
1answer
344 views

Formal Power Series — what's in it?

I have the following statement in a paper: Let $\Psi$ be the formal power series defined over the alphabet $\Omega$ and the log semiring by: $(\Psi, (a, b)) = -log(c((a,b)))$ for $(a,b) \in \Omega$, ...
3
votes
2answers
152 views

If it converges, how to show that power series converges to $f(x)$?

I had a very basic question. Suppose $f(x)$ is a function. And let us say it has a power series :- $$f(x) = \sum_{n=0}^\infty a_nx^n.$$ Suppose we are operating inside the region of convergence. ...
3
votes
1answer
586 views

Power Series With Bernoulli Numbers

The exercise reads "Express the power series for $\large \frac{z}{\sin (z)} = \frac{2 i z}{e^{iz} - e^{-iz}} $ in terms of Bernoulli numbers." I am given in a previous exercise that the Bernoulli ...
3
votes
3answers
2k views

Power (Laurent) Series of $\coth(x)$

I need some help to prove that the power series of $\coth x$ is: $$\frac{1}{x} + \frac{x}{3} - \frac{x^3}{45} + O(x^5) \ \ \ \ \ $$ I don't know how to derive this, should I divide the expansion of ...
3
votes
3answers
4k views

Finding the power series of $\arcsin x$

I'm trying to find the power series of $\arcsin x$. This is what I did so far: $(\arcsin x)'=\frac{1}{\sqrt{1-x^2}}$, so $\arcsin x=\int \sqrt{\sum_{n=0}^{\infty}x^{2n}}$. (for $|x|<1$) Any ...
2
votes
3answers
368 views

Simple Power Series Expansion for Problems similar to $f = (1 + \epsilon \,x)^{1/\epsilon}$

I was flicking through a book on perturbation methods and saw a simple question asking the reader to expand the following expression for $f$ in a power series (up to the first 2 terms): $f = (1 + ...
1
vote
3answers
97 views

Series expansion of $\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$?

How would I find the series expansion $\displaystyle\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$ so that it will turn into an infinite power series again??
1
vote
4answers
131 views

Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $

Evaluate: $$\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $$ I rewrote the sum as $$\sum_{n=0}^{\infty} \frac {1}{4^{8n-7}(8n-7)} - \sum_{n=0}^{\infty} \frac {1}{4^{8n-3}(8n-3)}$$ Now, I ...
8
votes
1answer
268 views

Does a convergent power series on a closed disk always converge uniformly?

If I have a power series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i} \in\mathbb{C}[[z]]$ with radius of convergence $r>0$ and I know that the series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
5
votes
2answers
173 views

What is the order of this pole?

$$f(z)=\frac 1{\cos(z^4)-1}$$ $z=0$ is a pole of $f$, and I believe that the Laurent series centred at $0$ is $-\frac 2{z^8}-\frac 16+...$, which looks like the pole is of order $8$, but why does ...
5
votes
3answers
1k views

Generalised Binomial Theorem Intuition

It was not until recently (why don't they teach it in secondary school?) that I've come across the Generalised Binomial Theorem, which from what I can tell is basically the same as the regular ...
5
votes
1answer
259 views

The net signed area between $t=0, y=0, t=x$, and $y=f(t)$

f(t) is continuous function.So I know that $\int _0^x {f(t) dt}=$ "The net signed area between $t=0, y=0, t=x$, and $y=f(t)$" And I can find the same result with endless small rectangulars areas ...
5
votes
1answer
939 views

How does a complex power series behave on the boundary of the disc of convergence?

Until recently, I thought my math teacher had said in his real analysis class that on the boundary of the disc of convergence, a complex power series has at least one point at which it does not ...
5
votes
3answers
398 views

A deceiving Taylor series

When we try to expand $$ \begin{align} f:&\mathbb R \to \mathbb R\\ &x \mapsto \begin{cases} \mathrm e^{-\large\frac 1{x^2}} &\Leftarrow x\neq 0\\ 0 &\Leftarrow x=0 ...
4
votes
2answers
359 views

Graphical interpretation of infinite power series?

Can someone please give me a graphical interpretation/sense of infinite power series? Some functions such as exponentials, sines, and cosines are infinite power series, but what does that mean and ...
4
votes
2answers
145 views

What's the background of this exercise?

I found this interesting exercise on a calculus book (Stewart) Let $$ u=1+\frac{x^3}{3!}+\frac{x^6}{6!}+\cdots $$ $$ v=x+\frac{x^4}{4!}+\frac{x^7}{7!}+\cdots $$ $$ ...
4
votes
1answer
2k views

Finding closed forms for $\sum n z^{n}$ and $\sum n^{2} z^{n}$

Using the identity $\frac{1}{1-z} = 1 + z + z^2 + \ldots$ for $|z| < 1$, find closed forms for the sums $\sum n z^n$ and $\sum n^2 z^n$. My solution: Because $\displaystyle1 + z + z^{2} + ...
4
votes
2answers
228 views

Proving the equivalence of a sum and a double integral

Based on "Certain Subclass of Starlike Functions" journal by Chun-Yi and Shi-Qiong Zhou in 2007 (Science Direct), I found difficulties to understand the proof in Theorem 3 where they have verified: ...
4
votes
3answers
253 views

Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$

Given the geometric series: $1 + x^2 + x^4 + x^6 + x^8 + \cdots$ We can recast it as: $S = 1 + x^2 \, (1 + x^2 + x^4 + x^6 + x^8 + \cdots)$, where $S = 1 + x^2 + x^4 + x^6 + x^8 + \cdots$. This ...
3
votes
4answers
75 views

First four terms of the power series of $f(z) = \frac{z}{e^z-1}$?

Attempt: $$ e^z = \sum_{n=0}^\infty \frac{z^n}{n!}$$ $$ e^z - 1 = \sum_{n=0}^\infty \frac{z^n}{n!} -1$$ $$ e^z - 1 = z\sum_{n=0}^\infty \frac{z^n}{(n+1)!} $$ Thus $$ \frac{z}{e^z-1} = ...
3
votes
1answer
298 views

How to properly translate the coefficients of a Taylor series?

Given a Taylor series $$f(z) = \sum_{k=0}^\infty c_k^{(a)}\frac{(z-a)^k}{k!}$$ of a meromorphic function $f$ in $\mathbb C$ (i.e. analytical except for a set of isolated points) around some value ...
3
votes
2answers
767 views

Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$?

I'm looking for an intuitive understanding instead of a formal proof. Thanks for the help.
3
votes
2answers
648 views

How to bound the truncation error for a Taylor polynomial approximation of $\tan(x)$

I am playing with Taylor series! I want to go beyond the basic text book examples ($\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, etc.) and try something different to improve my understanding. So I ...
3
votes
2answers
576 views

How to compute coefficients in Trinomial triangle at specific position?

I need to compute coefficients of $i$-th power of $x$ from simplifying of $$(x^2 + x + 1)^n$$ From that site i know about trinomial triangle. But how can compute coeficients of $i$-th element at ...
3
votes
5answers
416 views

Calculate sum of an infinite series

I have been struggling with this functional series. $$\sum_{n=1}^{\infty}{(-1)^{n-1}n^2x^n}$$ I need to calulate the sum.Any tips would be appreciated.
2
votes
1answer
36 views

limit of $a_n(x)=\frac{\sum_{k=0}^{t_n}x^k}{\sum_{k=n+1}^{t_n}x^k}$

Let $t>1$ and $[t_n]$ defines the integer part of the product $t.n$. I need to evaluate the sequence $$a_n(x)=\frac{\sum_{k=0}^{[t_n]}x^k}{\sum_{k=n+1}^{[t_n]}x^k}$$ is convergent or bounded as ...
2
votes
2answers
214 views

Evaluation morphisms of formal power series and nilpotent elements

Given a commutative ring $A$, and a finitely presented (associative) $A$-algebra $B$, show that a morphism of $A$-algebras $A[[x]] \longrightarrow B$ is given by evaluation at an nilpotent element $ ...
2
votes
0answers
168 views

Algorithm for reversion of power series?

Given a function $f(x)$ of the form: $$f(x) = x/(a_0x^0+a_1x^1+a_2x^2+a_3x^3+a_4x^4+a_5x^5+...a_nx^n)$$ Let $A$ be an arbitrary (any) infinite lower triangular matrix with ones in the diagonal: $$A ...
2
votes
1answer
394 views

Show that some $C^\infty$ real function is analytic

Consider the function $f:(a,b) \rightarrow \Bbb R $, which is $C^\infty$ class. It is required to show that if the function $f$ satisfies: $\forall_{n\in\Bbb N_+} \forall_{x\in(a,b)} f^{(n)}(x) \ge 0 ...
2
votes
2answers
998 views

Using Generating Functions to Solve Recursions

I have the recursion $A(n) = A(n-1) + n^2 - n$ with initial conditions $A(0) = 1$. I attempted to solve it using generating functions and I'm not quite sure I have it right, so I thought I might ask ...
2
votes
1answer
3k views

How to find the general solution of $(1+x^2)y''+2xy'-2y=0$. How to express by means of elementary functions?

Find the general solution of $$(1+x^2)y''+2xy'-2y=0$$ in terms of power series in $x$. Can you express this solution by means of elementary functions? I know that $y= ...
2
votes
2answers
561 views

Help on differential equation $y''-2\sin y'+3y=\cos x$

$y''-2\sin y'+3y=\cos x$ I'm trying to solve it by power series, but I just can't find the way to get $\sin y'$. Is there any special way to find it?
1
vote
2answers
38 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 ...
1
vote
1answer
54 views

Derivative of exp with definition of differentiability

Prove with the definition of differentiability that $\exp(z)$ is differentiable in $\mathbb C$ and $(\exp(z))' = \exp(z)$ for all $z \in \mathbb C.$ I tried: \begin{align*} \frac{\exp(z+h) - ...
1
vote
1answer
733 views

Is there a simple proof that a power series can be differentiated term by term?

I know that the derivative of the sum of a power series can be calculated by summing the derivatives of the terms, and that the resulting series has the same radius of convergence as the original. The ...
1
vote
1answer
176 views

Applications of higher powers of trigonometric functions

I am after a reference (book, papers etc) about the practical applications of trigonometric functions raised to higher powers. An example is one that I have been using in my own studies: $\cos^4 ...
1
vote
1answer
536 views

The coefficients in the inverse of unit lower triangular Toeplitz matrix

I have a question about the inverse of lower triangular Toeplitz matrix. There is a matrix $Q$ which can be infinite-dimensional. $$ Q=\left[\begin{array}{ccccc} 1\\ -k_{1} & 1\\ -k_{2} & ...
0
votes
1answer
163 views

What's the limit of coefficient ratio for a reciprocating power series?

I have a question about the coefficient in the inverse of the power series. Assume $$ f=1-\sum_{i=1}^{\infty}(ck_i)x^i, $$ where $c$ and $k_i$ are positive and $0<ck_i<1$ for any $i>0$. ...
0
votes
2answers
88 views

Trying to revert a series with problematic log term

I'm stuck on a problem which I'm not sure has a solution. I have the first few terms of a series I want to invert, $y(x)=\ln(x)+a_0+\frac{a_1}{x}+\frac{a_2}{x^2}+\cdots$ I know the inverse exists ...
-2
votes
1answer
43 views

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 ...
9
votes
4answers
321 views

Asymptotic behavior of $\sum\limits_{k=1}^n \frac{2^k}{k}$

I'm looking for an asymptotic equivalent of $$\sum_{0 < k \le n} \frac{2^k}{k}$$ as $n \to \infty$. A plausible candidate seems to be $\frac{2^{n+1}}{n+1}$ (WolframAlpha plot, and the intuitive ...
7
votes
1answer
98 views

How to derive this interesting identity for $\log(\sin(x))$ [duplicate]

I saw on SE that: $$\log(\sin x)=-\log(2)-\sum_{n=1}^{\infty}\frac{\cos(2nx)}{n} \phantom{a} (0<x<\pi)$$ This is an extremely useful identity, as it helps solve: $$\int_{0}^{\pi} ...
4
votes
2answers
143 views

the sum of $1-\frac{1}{5}+\frac{1}{9}-\frac{1}{13}+…$

I thought this was the real part of the series: $\sum_{n=0}^\infty \frac{i^n}{1+2n}$, with $i=\sqrt{-1}$. When taking the real part I am left with: $\sum_{n=0}^\infty \frac{\cos(n\pi/2)}{1+2n}$. I ...
4
votes
1answer
128 views

Convergence of the power series $\sum \left(\frac{n^n}{n!} x^n \right)$

Find the convergence radius of the serie $$\sum \frac{n^n}{n!}x^n $$ and analyze the absolute convergence and/or uniform. What I've done: It is easy to show that the radius of convergence of this ...
3
votes
4answers
400 views

Expansion of $(1-z)^{-m}$

Expand $(1-z)^{-m}$, $m$ a positive integer, in powers of $z$. Since $\dfrac{1}{1-z}=1+z+z^2+\ldots$, we can find $$\dfrac{1}{(1-z)^2} = (1+z+z^2+\ldots)(1+z+z^2+\ldots) = 1+2z+3z^2+\ldots.$$ ...
3
votes
3answers
940 views

General term of Taylor Series of $\sin x$ centered at $\pi/4$

What is the general term for a Taylor series of $\sin(x)$ centered at $\pi/4$? It should be $(-1)^{[??]} \times \sqrt{2}/2 \times \frac{(x-\pi/4)^n}{n!}$ What power is $(-1)$ supposed to be raised ...
3
votes
4answers
234 views

Show that $\displaystyle\lim_{n\to\infty} nx^n=0$ for $x\in[0,1)$.

Show that $\displaystyle\lim_{n\to\infty} nx^n=0$ for $x\in[0,1)$.