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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1answer
50 views

Does this complex power series converges on the unit circle?

Prove that the series $$ \sum_{n=1}^\infty \frac{(-1)^{[\sqrt n]}}{n}z^n$$ converges on $\partial B(0,1)$.Where $[x]$ implies the greatest integer that is not bigger than $x$. It is easy to prove ...
2
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0answers
65 views

Does$\sum_{n=1}^{\infty}a_nb_n\leq \sum_{n=1}^{\infty}a_n\sum_{n=1}^{\infty}b_n$ always hold true?

It is fairly obvious that $$\sum_{n=1}^{k}a_nb_n\leq \sum_{n=1}^{k}a_n\sum_{n=1}^{k}b_n$$ is true for all finite $k$ and positive $a_i,b_i$, but even if $$\sum_{n=1}^{\infty}a_nb_n$$ converges we ...
0
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1answer
54 views

Really Cool Power Series Coefficient Problem

Hi everyone :) We learnt what a power series is in class, but that coefficient thing is new. How do we find coefficients of power series using that equation? What do we do? If someone can help me ...
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2answers
77 views

Find the sum of the series: $\frac{1}{1*2} - \frac{1}{3*2^3} + \frac{1}{5*2^5} - \frac{1}{7*2^7}+\dots$?

$$\frac{1}{1*2} - \frac{1}{3*2^3} + \frac{1}{5*2^5} - \frac{1}{7*2^7}+\dots$$ I made a series to get $$\sum_{n=0}^\inf \frac{(-1)^n}{(1+2n)*2^{1+2n}}$$ but what series can it manipulate and simplify ...
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2answers
22 views

Find the sum of the series (manipulate to series first)

$$\sum_{n=1}^\inf (-1)^{n-1}\frac{3^n}{n5^n}$$ I recognize it must be one of $$e^x$$ or $$\frac1{1-x}$$ when expanded to series and must be manipulated to it but so far I made it to $$\sum_{n=1}^\inf ...
0
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1answer
25 views

Find the sum of the following series (manipulating it to a series)

$$\sum_{n=0}^{\infty} (-1)^n\frac{\pi^{4n}}{n!}$$ I'm unsure how to approach this any hints would be much appreciated!
2
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6answers
75 views

Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent?

Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent? I tried ratio test but didn't seem to work, and I also know that the limit goes to zero, but I can't say its convergence because ...
0
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1answer
28 views

A rational function f(x) has the following power series representation for the interval $-3<x<3$. $f(x) = x -\frac{x^2}{3}+\frac{x^3}{3^2}+…$

A rational function f(x) has the following power series representation for the interval $-3<x<3$. $f(x) = x - \frac{x^2}{3} + \frac{x^3}{3^2} +...$. Find a closed-form expression for f(x). Now, ...
0
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1answer
31 views

Obtain power series expansion of $\frac{1}{(1+x)(1-2x)}$

Obtain power series expansion of $\frac{1}{(1+x)(1-2x)}$ and give the general term and radius of convergence.
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2answers
39 views

Number of terms to estimate the integral within the indicated accuracy using series!

$$\int_{0}^{0.4}\sqrt{1+x^4} dx$$ with |Error| less than or equal to $$\frac{0.4^9}{72}$$ So far i have broken down the sqrt to $$(1+x^4)^{1/2}$$ then made it into the series $$\sum_{n=0}^\infty ...
1
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1answer
33 views

Prove that for a power series function that is constantly zero, that the coefficients are zero

Suppose that power series function $a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots$ is constantly zero on a bounded non-empty open interval $I$, which may or may not contain $0$. Prove that $a_j ...
2
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2answers
52 views

Determine the first three non-zero terms in the Taylor polynomial approximation for the initial value problem: $y''+\sin(y)=0$

Having trouble understanding how to solve this problem. Did I at least set it up correctly? $y''+\sin(y)=0,\;y(0)=1,\;y'(0)=0$ So assuming $y(x)=\sum_{n=0}^{\infty}a_nx^n$ then ...
1
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1answer
31 views

How to use a generating function to work out an infinite sum

I have the infinite sums: $$\sum_{k=0}^\infty k^2a^k \quad \text{and}\quad \sum_{k=0}^\infty ka^k$$ where, $\left\lvert a \right\rvert<1$. I was able to find the answers to the infite sums here, ...
0
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2answers
44 views

When is a finite sum of powers of non-integer a rational number? [closed]

Concretely, is there $ b \in \mathbb R, n,k \in \mathbb N $ such that $ \sum_{i = n}^{n+k} b^i \in \mathbb Q$ ?
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2answers
51 views

Could the sum of powers of non-integers result in a whole number? [closed]

Concretely, is there a $ b \in \mathbb R $ such that $ \sum_{i \in I \subset \mathbb N} b^i \in \mathbb W$ ?
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5answers
57 views

Find the general solution of $y''+y=0$

This is brand new material that I'm trying to learn and it says: To find a power series solution about the point $x = 0$, we write $$y(x)=\sum_{n=0}^{\infty}a_nx^n$$ So then I differentiate: ...
1
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1answer
80 views

For a power series function that is constantly zero prove that the coefficients are zero

Suppose that power series function $a_0 + a_1x + a_2x^2 + a_3x^3 + ...$ is constantly zero on a bounded non-empty open interval $I$, which may or may not contain $0$. Prove that $a_j = ...
1
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1answer
21 views

Convergence of the power series

Does the power series $\sum_{n=1}(1 - 1/sqrt(n))^n $ converge or diverge? I tried root test but it doesn't work. Which convergence test should I try to solve the problem?
0
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1answer
27 views

Finding the largest domain on which a series expansion converges

I need to find the largest domain on which the series expansion of $f(z) = -\frac{1}{z-\pi/2}-\frac{1}{z+\pi/2} = 2*\sum_{k=1}^{\infty} (\frac{2}{\pi})^{2*k}z^{2k-1}$. I tried using the ratio test, ...
2
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2answers
44 views

Uniform convergence of power series $\sum_{n=1}^\infty \frac {x^n}{(n+1)(n+2)}$

Prove the uniform convergence of power series $$\sum_{n=1}^\infty \frac {x^n}{(n+1)(n+2)}$$ on the closed interval $[-1,1]$. The radius of convergence $$R = \lim_{n\to\infty} | \frac{a_n}{a_{n+1}}| ...
0
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2answers
28 views

Find the Laurent series of $\frac{1}{1+x^2}$ centered around $i$

I am having trouble with finding the Laurent series of the following function, centered around $i$: $$f(x) = \frac{1}{1+x^2} .$$ I tried transforming this into a form that would be appropriate ...
1
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1answer
106 views

Using power series to evaluate $\int x^2\sin(x^2)dx$

Since I know the series of $\sin(x^2)$ is $$\sum_{n=0}^\infty \frac{(-1)^nx^{4n+2}}{(2n+1)!}$$ then do I multiply $x^2$ into this to get $$\sum_{n=0}^\infty \frac{(-1)^nx^{4n+4}}{(2n+1)!}$$ and then ...
2
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6answers
66 views

Sum of a series to an exact answer

I am trying to work out what this series evaluates to: $$\sum_{i=1}^\infty i(1-k){k^{i-1}} $$ where k is a constant such that 0 < k < 1. To figure this out I expand the brackets to get: ...
2
votes
2answers
34 views

Determine the radius of convergence of the series $\sum \limits_{n=0}^\infty (10^{−n} + 10^n)x^n$

Determine the radius of convergence, R, of the power series $\sum \limits_{n=0}^\infty a_nx^n$ where $a_n = 10^{−n} + 10^n$. I know the series diverges and the radius of convergence is 0 but how do I ...
1
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1answer
15 views

Are all series in the elementary Ramanujan class R = 1 non-summable by analytic continuation of Dirichlet series?

We say that a series $\sum_{n=1}^\infty a_n$ and the corresponding power series $f(x)=\sum_{n=1}^\infty a_nx^n$ belong to the Ramanujan class $R=1$ if $g(x)=f(x)-f(x^2)$ is Abel summable at $x=1$ ...
1
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1answer
37 views

Matrix series convergence

Suppose we have the Maclaurin series of a function $f$, and it converges in a radius $R$. Then suppose we define a matrix argument to the function in a similar manner to the exponential definition of ...
6
votes
4answers
108 views

Power series solution for ODE

The ODE I have is $$y'(x)+e^{y(x)}+\frac{e^x-e^{-x}}{4}=0, \hspace{0.2cm} y(0)=0$$ I want to determine the first five terms (coefficients $a_0,\ldots, a_5$) of the power series solution ...
1
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1answer
21 views

Equality of coefficients in this power sereis

Hei, I'm trying to check if the equalit of power series (valid in some open neighbourhood of $z = 0$): $$ \sum_{i=1}^\infty z^ic_i + \sum_{i=1}^\infty \overline{z^ic}_i = \sum_{i=1}^\infty z^id_i + ...
1
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2answers
67 views

The integral of $ \sin(x^2) $

The actual problem is: Find the first four nonzero terms and the general term of the Maclaurin series for $g(x) = \int\sin(x^2)dx $ with $g(0) = 1$. How do you find the integral of $\sin(x^2)? $ I ...
0
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0answers
32 views

Finding radius of convergenc of a power series expansion

I am trying to solve the following problem but have no idea how to approach it. I was wondering if anyone could give me any hint or suggestion. Suppose $f(z)$ is analytic at $z=0$ and satisfies $f(z) ...
0
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2answers
25 views

Fining the radius of convergence of $\sum_{k=1} ^{\infty} \frac{2^k*z^{2k}}{k^2+k}$

I have been trying to get the radius of convergence of this expression $\sum_{k=1} ^{\infty} \frac{2^k*z^{2k}}{k^2+k}$ by using the ratio test. However I keep getting $\frac{1}{2}$ and the answers say ...
4
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1answer
81 views

How to evaluate the series $\sum_{i,j,k=0}^{\infty}\left(\frac{(i+j+k)!}{i!j!k!}\right)^2x^{-i-j-k} $?

Suppose the series $$ \Gamma (x) =\sum_{i,j,k=0}^{\infty}\frac{((i+j+k)!)^2}{(i!)^2(j!)^2(k!)^2}x^{-i-j-k} $$ How to evaluate it? It is claimed that for $x <3$ this function converges to elliptic ...
1
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0answers
21 views

Borel-Laplace transform

I have some questions about Borel-Laplace transforms: I can not find any good references explaining in details all what follows: Let $f(z)=\sum_{n\ge0}\limits a_nz^n$ be an entire function on ...
2
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1answer
45 views

Find the Maclaurin series for $\ln(2-x)$

A little unsure if the result I got makes sense, so I want to ask here to be sure I am not doing something very silly. The Maclaurin series is given by ...
1
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2answers
51 views

Prove $\frac{d}{dx} \int_x^{x^2}\ \frac{\sin t}{t} dt = \frac{2\sin x^2 - \sin x}{x}$

Check whether the following is true: $$\frac{d}{dx} \int_x^{x^2}\ \frac{\sin t}{t} dt = \frac{2\sin x^2 - \sin x}{x}$$ . If not true then prove it wrong. I know how to evaluate $$\int\frac{\sin ...
0
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1answer
27 views

Use partial fractions to write $\frac{1}{x^2 + x – 1}$ as $\frac{A_1}{x – α_1} + \frac{A_2}{x – α_2}$ and write it as a power series

Find the roots $α_1$, $α_2$ of $x^2 + x – 1$ and use partial fractions to write $\frac{1}{x^2 + x – 1}$ as $\frac{A_1}{x – α_1} + \frac{A_2}{x – α_2}$ , for suitable $A_1, A_2$. Using the power series ...
1
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0answers
30 views

I'm looking to find a power series for a solution f where $zf''(z)+f(z)=0$

I'm looking to find a power series for a solution f where $zf''(z)+f(z)=0$ and $f(0)=0$ and $f'(0)=1$ and with the assumption $\sum_{k=1}^∞{a_k z^k}=0$ So far I have done this: $\sum_{k=1}^∞{k^2 ...
3
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1answer
23 views

Invertible elements of a power series ring

Let $F$ be a field and $F[[x]]$ be the power series ring with coefficients in $F$. It seems if $\alpha, \beta \in F[[x]]$, with $\alpha^{-1} = \beta$, then all coefficients of the product $\alpha * ...
3
votes
1answer
39 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
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1answer
40 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
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1answer
22 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
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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
33 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
48 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
34 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
63 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
35 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
44 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
91 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 ...