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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3
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1answer
38 views

Radii of convergence for complex series

I need to find the radii of convergence for these series: $1. \sum_{n=1}^\infty (2+(-1)^n)^n z^{2n}$ $2. \sum_{n=1}^\infty (n+a^n)z^n, a \in C $ $3. \sum_{n=1}^\infty 2^n z^{n!}$ Starting with ...
2
votes
1answer
39 views

A series whose terms are the products of terms of a geometric and a power series

Consider this summation $$ \sum_{i=1}^{\infty}\frac{1}{i^ab^i} $$ where $a$ and $b$ are greater than $1$ It can be upper bounded by the geometric series ...
1
vote
1answer
19 views

How does $\sum_{t=0}^\infty(1-\frac2n)^t\frac{e^{-n\lambda }(n\lambda)^t}{t!}=e^{-n\lambda}\sum_{t=0}^\infty \frac{[\lambda(n-2)]^t}{t!}.$

How does $$\sum_{t=0}^\infty(1-\frac2n)^t\frac{e^{-n\lambda }(n\lambda)^t}{t!}=e^{-n\lambda}\sum_{t=0}^\infty \frac{[\lambda(n-2)]^t}{t!}.$$All I see is $e^{-n\lambda}$ getting pulled out.
0
votes
1answer
68 views

Write $\sum_{n=1}^\infty a_nx^{n-1} = 1+x+2x^2+3x^3+5x^4+8x^5+13x^6+21x^7+34x^8+55x^9+89x^{10}+… $ as a power series [closed]

Let $\sum_{n=1}^\infty a_nx^{n-1} = 1+x+2x^2+3x^3+5x^4+8x^5+13x^6+21x^7+34x^8+55x^9+89x^{10}+\ldots $ Use the ratio test to prove that f(x) converges if |x|< $\frac{1}{2}$ . Edit: $a_n$ in this ...
0
votes
1answer
32 views

What happens to the Radius of Convergence of a Geometric Series when you take its anti-derivative?

I've just learned that you can sometimes turn functions into infinite geometric series as long as the independent variable is restricted to an interval of convergence such that the abs value of the ...
2
votes
1answer
46 views

What's wrong with my infinite series expansion for $\log(x)$?

Here, log is natural log. Looking at $f(x)=\frac{1}{x}$, I tried to put $f(x)$ in the form $\frac{a}{1-r}$ that an infinite geometric series $\sum_{n=0}^\infty (a \cdot r^n)$ converges to when $\mid ...
0
votes
2answers
33 views

Expand $f(z)=\frac{1}{z^2(z-i)}$ in 2 different Laurent expansions around $z=i$ and tell where each converges.

My attempt: $$f(z)=\frac{1}{z^2(z-i)}$$ $$\frac{1}{z^2(z-i)}=\frac{Az+B}{z^2}+\frac{C}{(z-i)}$$ Solving for the unknown constants yields $$A=1$$ $$B=i$$ $$C=-1$$ Thus, $$f(z)=\frac{z+i}{z^2} - ...
0
votes
2answers
62 views

Show that $\exp(-\lambda x) \cdot\exp(\lambda x)=1$ using the power series

Let $A$ be a commutative Banach algebra. Consider the exponential function $$\exp(\lambda x) = \sum_{n=1}^\infty\frac{(\lambda x)^n}{n!},$$ where $x \in A$ and $\lambda \in \mathbb C$. We can easily ...
0
votes
2answers
34 views

Confusion about Power Series Representation

I have to find a power series representation for $f(x) = \frac{x-1}{x+2}$. In rearranging the function so as to attain a form suitable for representation as a power series I get $$(x-1) * ...
1
vote
1answer
43 views

Geometric series $ar^n$ where $n \ne 1,2,3,4 \cdots$

The probability of rolling a seven on two dice is as follows $$p=1/6+1/6(5/6)^2+1/6(5/6)^4 + \cdots$$ what is the probability of rolling a $7$? Is there an advantage to rolling first? My attempt at ...
1
vote
2answers
29 views

Is my answer to this power series representation problem right?

Find power series representation of the function $f(x) = \frac{3}{x+2}$ \begin{align*}f(x) = \left(\frac{3}{x}\right)\frac{1}{1-\left(-\frac{2}{x}\right)} = \left(\frac{3}{x}\right) ...
0
votes
1answer
90 views

Power series centered at $0$ which converge to $\sinh$

Determine all power series centered at $0$ (i.e. equal to $\sum_{n=0}^\infty a_n x^n$)which converge to the hyperbolic sine $\sinh: \mathbb{C} \to \mathbb{C}, z \mapsto \frac{\sin(iz)}{i} $. My ...
1
vote
1answer
94 views

Proof for this 'power triangle' [closed]

Is there a proof for this triangle Using factorial as a difference of powers? The first row is every consecutive integer raised to the power of n(5 here), but when you write the difference of them, ...
2
votes
0answers
22 views

Series exapnsion of the sum of the $3n^3$-th powers of the first n positive integers

Let $S_{n,m}$ be the sum of the m-th powers of the first n positive integers: $$S_{n,m}=\sum_{k=1}^n k^m=\sum_{k=1}^n (n-k+1)^m\tag{1}$$ If $m\in\mathbb{N}$, and $m<<n$, we can use the ...
2
votes
1answer
48 views

endpoints convergence after integrating/mutiplying /subtracting power series

when we multiply a power series that converges for all values of $x$ by another power series of interval of convergence $(-1,1]$, then the new interval of convergence is the intersection of the 2 ...
1
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1answer
39 views

Find the Laurent series for $f(z)=\frac{1}{z^2-3z+2}$ in the region of $0<|z-2|<1$

My attempt: $$f(z)=\frac{1}{z^2-3z+2}$$ $$=\frac{1}{(z-2)(z-1)}=\frac{A}{(z-2)}+\frac{B}{(z-1)}$$ After finding common denominator, equating the numerators, and letting $z=0$, we get: ...
2
votes
1answer
39 views

The sum of k times the kth power of a is given analytically by?

I was wondering how would someone derive (Not prove) the result in terms of n and a of the following sum: $$\sum_{k=1}^n ka^k$$ My question basically is, given that summation how would you tackle ...
0
votes
1answer
44 views

Uniform convergence of $\sum_{n\geq1}{z^n \over n}$ over a circular arc

As a follow-up to this question Find all the points which makes the series normal converges and uniformly converges, I'm wondering if the series of function $\sum_{n\geq1}{z^n \over n}$ converges ...
0
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0answers
19 views

Asymptotic Expansion for Function with an Embedded Integral [duplicate]

So I'm trying to find the asymptotic expansion as $x \to \infty$ of: $$f(x)=\frac{1}{\bigg[A-\int_{x_0}^x\frac{\lambda^y}{\Gamma(y+1)}dy\bigg]^{\frac{1}{\alpha}}}$$ where $x_0>0$ and $\alpha>0$ ...
2
votes
2answers
40 views

Find the Laurent series for $\frac{\cos z}{z^2}$ centered at $z=0$

1st attempt: The power series expansion for $\cos z$ is: $$f(z)=1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}...$$ Dividing by $z^2$ gives: $$f(z) = ...
0
votes
1answer
38 views

Radius of convergence in complex analysis

We have an analytic function $f$ in some open ball $B(0,1)$. Assume radius of convergence of the power series of $f$ at $0$ is 1. Consider the taylor series of $f$ at $1/2$. Then its radius of ...
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2answers
69 views

Find all the points which makes the series normal converges and uniformly converges

I'm learning "Complex Analysis", section Series and Convergence, and I got stuck on this problem (actually, just a small part of this problem): Find all the values of $z$ which makes this series ...
0
votes
1answer
26 views

Functions as a Power Series

So, say you had $\dfrac{(-4)^n}{-4}$ Would that be equivalent to $(-1)^{n-1} \times 4^{n-1}$? If so, how come it isn't $1^{n-1}$ (or just $1$) ?
1
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2answers
47 views

Finding the power series of $\frac{1}{z^2}$

I need help finding the power series for this function around $z=1$: $$f(z)=\frac{1}{z^2}$$ My attempt to solve this: We know: $\frac{1}{1-x}=\sum_{n=0}^\infty x^n$, so I tried substituting ...
0
votes
1answer
51 views

Sum of the series $\sum_{n=0}^{\infty} \frac{1}{4^n(2n+1)}$

I am having difficulty with calculating this sum. By the Ratio Test, the series converges and on the solution key of the last year's exam it written that the sum = $\ln 3$. I tried known Maclaurin ...
2
votes
3answers
38 views

Radius of convergence involving $z^{n^2}$

Consider the following complex power series: $\displaystyle\sum_{n=1}^\infty a_nz^{n^2}$ where , $\displaystyle a_n = \frac{1}{n!}$. My approach is that by ratio test $$\lim\limits_{n\to\infty} ...
0
votes
1answer
31 views

Power series $\sum^{\infty}_{n=0}a_n(x-x_0)^n$ - absolutely convergence

Why if a power series $\displaystyle\sum^{\infty}_{n=0}a_n(x-x_0)^n$ converges in $x=x_1$, then it it absolutely convergent when $|x-x_0|<|x_1-x_0|$? and if the power series diverges in $x=x_1$, ...
1
vote
2answers
58 views

Find a rational number that agrees with this in its first four decimals [closed]

I'm studying power series now and I'm asked to find a rational number that agrees with: $$\int_0^{1/2} \sqrt[3]{1+x^2}dx$$
0
votes
1answer
32 views

Solving a non-homogeneous differential equation via series solution

I have been set a question which asks me to solve: $$y''-2y=x^2e^{x^2}$$ by using a power series method. In trying to do it by brute force I end up with an non-homogeneous recurrence relation which ...
0
votes
1answer
21 views

Alternate (approximate) form for Hypergeometric function 1F1(0.5, 1.5, -x)

I have the following Hypergeometric function of the first kind: $_{1}F_1(\frac{1}{2}, \frac{3}{2}, -x)$ where $x$ is not negative. This function can also be written as the following series: ...
0
votes
1answer
59 views

How to compute $\sum_{n=1}^{+\infty}\frac{1}{n!2^{3n}}$?

I want to find the following sum: $$ \sum_{n=1}^{+\infty}\frac{1}{n!2^{3n}} $$ However I am not sure that the following computation is true: $$ ...
2
votes
2answers
40 views

Definite Integral, use a power series.

Use a power series to approximate the definite integral, I, to six decimal places. I tried to definite integral, but the answer is wrong. Where did I make a mistake? $$\int_{0}^{0.2} \ln(1+x^4)\ dx ...
2
votes
0answers
39 views

Writing $\frac{1}{1 + w + w^2}$ as a power series and finding the ROC

I have to write the following: $\frac{1}{1 + w + w^2}$ as a power series: $$\sum_{n=0}^{\infty}{a_nw^n}$$ and find the radius of convergence of the series (in the complex plane). Obviously you can ...
3
votes
1answer
70 views

Show that each of the coefficients of a complex power series are real

Suppose that $g(z) = \sum c_nz^n$ has radius of convergence $R > 0$ and that $g(\frac{1}{m})$ is real for all $m > \frac{1}{R}$. Show that each $c_n$ is real. I know that if $R>0$ ...
1
vote
1answer
38 views

Determining a complex function represented as a series

Find the domain $D$ of the function $ f(z) = \sum (-1)^n(z-1)^n - \sum i^{n-1} (z-i)^n $ Then determine $f(z)$, for all $z \in D$ I figured for the first part of the function i could ...
1
vote
0answers
73 views

Asymptotic Expansion for a Function involving a Weird Integral

So I'm trying to find the asymptotic expansion as $x \to \infty$ of $$f(x)=\frac{1}{\bigg[A-\int \frac{\lambda^x}{\Gamma(x+1)}dx\bigg]^\frac{1}{\alpha}}$$ Note that $\lambda>0$ and $\alpha>0$. ...
1
vote
2answers
69 views

Series expansion of $ \frac{x}{\ln (1+x)}$ [closed]

What are coefficients in the expansion of series for $$ \frac{x}{\ln (1+x)} = \sum_{n=0}^\infty A_n \frac{x^n}{n!}?$$ Do they have a name?
1
vote
1answer
52 views

Does the Abel sum 1 - 1 + 1 - 1 + … = 1/2 imply $\eta(0)=1/2$?

If $\sum_{n=1}^\infty a_n$ is Abel summable to $A$, then necessarily $\sum_{n=1}^\infty a_n n^{-s}$ has a finite abscissa of convergence and can be analytically continued to a function $F(s)$ on a ...
4
votes
1answer
72 views

Proving complex trigonometric identity using power series

Prove $2$cos$^2(z) = 1+$cos$(2z)$ using power series. I know that cos$(z) = \sum (-1)^n\frac{z^{2n}}{(2n)!}$ I also know that if $a(z) = \sum a_nz^n$ and $b(z) = \sum b_nz^n$ then $a(z)b(z) ...
0
votes
1answer
15 views

Find (with proof) the domain of each of the following power series functions

Four functions here I am not quite sure how to go about doing this. I feel I am missing something very easy. Can someone please help? I am preparing for an exam and this is important.
1
vote
0answers
33 views

Power Series Solution to ODE

Trying to find the general solution of the following equation: $$2x(1+2x^2)y''-y'-24xy=0 $$ using: $$y(x)=\sum_{n=0}^\infty a_nx^{n+r}$$ r and $a_n$ are constants I get so far as the recurrence ...
3
votes
3answers
109 views

Calculate $\sum\limits_{i=2}^\infty\sum\limits_{j=2}^\infty \frac{1}{j^i}$

Calculate $\sum\limits_{i=2}^\infty\sum\limits_{j=2}^\infty \frac{1}{j^i}$. I am trying to figure out how to calculate this. I know it must be ...
1
vote
1answer
44 views

Formula for a geometric series weighted by binomial coefficients (sum over the upper index):$\sum_{i=0}^L {n+i\choose n}\ x^i =\ ?$

The binomial sum is $$\sum\limits_{i=0}^n {n\choose i}\ x^i = (1+x)^n,$$ where $\displaystyle{n\choose i}=\frac{n!}{(n-i)!i!}.$ Is there a corresponding formula when you sum over the upper index of ...
1
vote
1answer
32 views

Want to check that $\sum_{j=0}^{k-1}w^{ jm}=0$, $m\not\equiv 0 \pmod{k}$ where $w=e^{2\pi i/k}$

If $f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$, then $$ \sum_{n=0}^{\infty}a_{kn+m}x^{kn+m}=\frac{1}{k}\sum_{j=0}^{k-1}w^{-jm}f(w^j x) \tag{1},$$ where $w=e^{2\pi i/k}$ is a primitive $k$th root of ...
0
votes
0answers
42 views

How to find the Laurent series for $\frac{1}{(z-1)(z-2)}$?

How to find the Laurent series for $$\frac{1}{(z-1)(z-2)}$$ on $|z|<2$ Here is what I did so far: $$\frac{1}{(z-1)(z-2)}=\frac{1}{(z-2)}-\frac{1}{(z-1)}$$ now ...
3
votes
1answer
44 views

Why does $\frac{1}{1-z}=-\sum_{n=0}^{\infty} \frac{1}{z^n}$ if $|z|>1$

Why does $$\frac{1}{1-z}=-\sum_{n=0}^{\infty} \frac{1}{z^n}$$ if $|z|>1$ I know the case for $|z|<1$ and if $|z|>1 \implies |1/z|<1$ so $$\frac{1}{1-\frac{1}{z}}=\sum_{n=0}^{\infty} ...
3
votes
1answer
47 views

Closed form for $\prod_{n=0}^\infty (1-z^{2^n})$

Is there a closed form for the product $$f(z) = \prod_{n=0}^\infty 1-z^{2^n}$$ either as a formal power series or as an analytic function in the disk $|z| < 1$? It's not hard to see that Taylor ...
1
vote
3answers
56 views

Finding the value of $\sum_{k=1}^{\infty} \frac{x^{4k-3}}{4k-3}$

This is my power series: $$\sum_{k=1}^{\infty} \frac{x^{4k-3}}{4k-3}$$ I need to find the sum of it. Unfortunately, I have kind of no idea how to do it. I think I need to substitute with ...
0
votes
0answers
21 views

A linear operator that extends the summation of Dirichlet series

Consider a vector space $\mathcal{V}$, a linear operator $L$ and a vector subspace $\mathcal{A}$ such that for all $x\in\mathcal{A}$ $Lx\in\mathcal{A}$ and for a number $R\neq0,1$ $R^{-1}$ is not a ...
1
vote
1answer
13 views

Prove that if there exists $c,M>0$ such that $|a_nc^n|\leq M\forall n$, then $(-c,c)\subseteq\operatorname{Conv}(\sum a_nx^n)$

Prove that if there exists $c,M>0$ such that $|a_nc^n|\leq M\,\forall n$, then $(-c,c)\subseteq \operatorname{Conv}(\sum a_nx^n)$. I tried to use the ratio test to, but I only get that $c\limsup ...