Tagged Questions
0
votes
2answers
19 views
Develop the next function:$f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0)$
Develop the next function:$\displaystyle f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0).$
For the first part:
$\displaystyle\frac ...
1
vote
2answers
56 views
Identity with Bernoulli numbers: $\sum\limits_{k=1}^{n}k^p=\frac{1}{p+1}\sum\limits_{j=0}^{p}\binom{p+1}{j}B_j n^{p+1-j}$
How I can prove that
$$\sum_{k=1}^{n}k^p=\frac{1}{p+1}\sum_{j=0}^{p}\binom{p+1}{j}B_j n^{p+1-j},$$
where $B_j$ is the $j$th Bernoulli number?
I hope to find the answer. Thanks for help.
1
vote
2answers
41 views
Finding the $x^n$ coefficient of the power series $\sum\limits_{n=0}^\infty\frac{x^{2n+3}}{n!}$
I have a practice test question that asks:
Given the following Maclaurin series representation, $$\sum\limits_{n=0}^\infty\frac{x^{2n+3}}{n!}$$ what is the coefficient of $x^n$?
I have the ...
2
votes
1answer
38 views
Radius of convergence of a power series with Bernoulli numbers
Say, we use the definition:
Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$
and then derive power series representations of the ...
2
votes
2answers
58 views
How does one get the Bernoulli numbers via the generating function?
Here is the definition:
Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$
I've tried to naively expand $\frac{x}{e^x-1}$ around ...
1
vote
1answer
33 views
Points around which one expands and the radiuses of convergence
I'm trying to make sense of the following passage:
Let $f(x)=\frac{1}{x+1}$ and $R_0$ the radius of convergence of the Taylor series of $f$ around $x_0=0$, analogously: $R_1$ — around ...
2
votes
1answer
47 views
Radius of convergence of $\sum_{n=1}^{\infty} { (n \sin{\frac{1}{n}})^{n} x^n } $
We need to calculate the radius of convergence $R$ of:
$$\sum_{n=1}^{\infty} {\left(n \sin{\frac{1}{n}}\right)^{n} x^n }.$$
Here's what I did:
$$ \lim_{n\to\infty} { \left| ...
3
votes
2answers
73 views
May I use the triangle inequality for infinite series?
I have to prove the following statement:
Let $\lim_{n\to \infty}r_n=0$. Show that $\forall\varepsilon>0 \ \ \ \exists \, n_0 \in \mathbb N \ \ \ \forall x \in(-1,1):$
$$\left\lvert ...
0
votes
2answers
59 views
Show that cosh(2) is between two values.
I'm reviewing for exams and this question has got me stumped:
Show that:
$3\dfrac{2}{3} \leq \cosh(2) \leq 3\dfrac{2}{3} + 0.1$
I've determined the series form of cosh(x) to be:
...
2
votes
1answer
56 views
Why do power series converge to a function symmetrically?
Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$?
The selected answer to the above question says that for a a power series, the interval of convergence for the ...
2
votes
2answers
133 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.
0
votes
2answers
46 views
Taylor series of $f(x^2)$
If you know the taylor series for $f(x)$ can you find the taylor series for $f(x^2)$ by letting $x = x^2$? The taylor series in question is $\cos(x^2)$
I know the taylor series for $\cos(x)$ is ...
1
vote
1answer
63 views
First few coefficients of a power series
The function $f(x) =\frac{5}{1+16x^2}$ is represented by the power series $$\sum_{n=0}^{\infty} c_nx^n$$
I'm supposed to find the first few coefficients of the power series, and these are the answers ...
1
vote
1answer
48 views
Why do Maclaurin series approximate a function for negative domain values?
A common analogy used as an intuitive explanation for a Maclaurin series is that of a car. If you know the position, velocity, acceleration, jerk etc. of a car at time zero, you are able to predict ...
2
votes
4answers
37 views
Radius convergence of power series
How would I go about finding the radius of convergence using a limit ratio test? Can I get a hint for this one?
$\displaystyle\sum_{j=1}^\infty\frac{(jx)^j}{j!}$
1
vote
2answers
32 views
Lagrange remainder to approximate $3^{2.1}$ less than 0.1
How do I solve this problem:
Use the appropriate Taylor polynomial $P_n(x,c)$ to estimate $3^{2.1}$ with error less than $0.1$, given $\ln 3$ is about $1.099$.
I understand that the remainder ...
1
vote
1answer
41 views
Find taylor polynomial that approximates e^x with accuracy at least 1.
Find Taylor polynomial at $x=0$ which approximates $e^x$ with accuracy at least $1$ for each $x \in [-2,2]$.
I dont undestand these questions that involve the $n^{th}$ remainder. I know I need to ...
1
vote
2answers
31 views
Radius of convergence in a series. Ratio test.
I am having a hard time with this question.
$$\sum_{k=0}^{\infty} \frac{-(1)^k (4^k -3)x^{2k}}{k^4+3}$$
I used the ratio test and got stuck here:
$$x^2 \lim_{k\to\infty} \frac ...
6
votes
1answer
68 views
A question regarding representation of a function as a power series
I'm trying to help my brother with a calculus problem related to the representation of a function as a power series. The task is to find what power series is represented by the following function, and ...
2
votes
2answers
48 views
Solving limit by substituting a power series
I dont understand why I am getting 2 and the textbook says it is -2.
$$\lim_{x\to 0} \frac{1-e^x}{\sqrt{1+x}-1}$$
I subbed the power series for $e^x$ and $(1+x)^{1/2}$ then got rid of the $1$ on top ...
2
votes
1answer
114 views
Convergence of a power series function
Consider the following differential equation:
$$w''(x)+p(x)w'(x)+q(x)w(x)=r(x)$$
with the initial condition of $w(0)=w_0,\ w'(0)=w_1$, and
$$w_{n+2}=\frac{r_{n+2}-(n+1)p_0w_{n+1}-\sum_{k=0}^n w_k ...
1
vote
2answers
88 views
Finding the Maclaurin series
Find the Maclaurin series for $f(x)=(x^2+4)e^{2x}$ and use it to calculate the 1000th derivative of $f(x)$ at $x=0$.
Is it possible to just find the Maclaurin series for $e^{2x}$ and then multiply it ...
5
votes
3answers
64 views
Representing Functions as Power Series
Rewrite $$f(x)=(1+x)/(1-x)^2$$ as a power series.
Work thus far:
I separated it into two parts:
$$1/(1-x)^2 + x/(1-x)^2$$
I realize that the first expression is the derivative of $1/(1-x)$ and ...
2
votes
1answer
31 views
Specific question about the consequence of composing power series
Please bear with my possible abuse of notation/terminology. Consider the power-series composition f(g(x)). If g's range lies within f's interval of convergence, and if series g has a constant term 0, ...
9
votes
3answers
154 views
Can the the radius of convergence increase due to composition of two power series?
When composing power series, is the radius of convergence the minimum of that of the individual series, or is it like for multiplication and addition of power series where the resultant radius of ...
1
vote
1answer
82 views
What is the radius of convergence?
$\displaystyle\sum_{n=1}^{\infty}x^{2n-1}/a_{n}$
What is the radius of convergence?
Ps: I found that $\limsup|1/a_{n}|^{1/n}=1/6$
But I am confused because of $x^{2n-1}$
What is the radius of ...
1
vote
1answer
18 views
What values does the function $Z(y)$ have at various interval?
When $y\leq0$; $H(y)=0$
When $y>0$; $H(y)=e^{-\dfrac1y}$
What values does the function $Z(y)$ have at various interval? Where $Z(y)=H(1-y)(1+y)$
Please show this!
3
votes
2answers
70 views
Find the radius of the series
$$\sum_{n=1}^{\infty}\frac{x^{n}}{n^{2}{(5+\cos(n\pi/3))^{n}}}$$
What is the radius of the convergence of the series?
Please show clearly and help me how to solve this. Thank you!
I know the ...
1
vote
2answers
111 views
Show that the radius of convergence of the power series is at least 1
If the coefficients ${a_i}$ of a power series $\displaystyle\sum_{i=0}^{\infty}a_{i}x^{i}$ form a bounded sequence show that the radius of convergence of the power series is at least $1$
How to solve ...
0
votes
1answer
27 views
Power Series Convergence
I have the series $$\sum_{n=1}^\infty2^{\frac{1}{n}}$$ and I want to test for convergence. So I wrote the series as
$$\sum_{n=1}^\infty2\cdot2^{\frac{1}{n-1}}$$
But I'm not sure how to put this into ...
0
votes
2answers
29 views
What does $\sum_{n=0}^{\infty} (-1)^n .\frac{x^{2n+1}}{2n+1}$ converge to at x= 1 and x = -1
This is what I did.
$\sum_{n=0}^{\infty} (-1)^n .\frac{1^{2n+1}}{2n+1}=\sum_{n=0}^{\infty}(-1)^n .\frac{1}{2n+1}$
Now I broke it up to positive and negative.
$\sum_{n=0}^{\infty}(-1)^n ...
1
vote
2answers
90 views
What is the sum of this power series?
This is the series:
$$ \sum_{n=1}^\infty \frac{x^{2n}}{(2+\sqrt{2})^n} $$
My problem is that I don't know how to rid of that $ (2+\sqrt{2})^{-n} $.
1
vote
1answer
122 views
Fibonacci Generating Function of a Complex Variable
So I'm doing work on the Fibonacci Numbers, and I came across this problem for the generating function
for the recursive fibonacci numbers.
I have two questions:
1. Why is it useful to use a ...
0
votes
1answer
102 views
Laurent series of $$ g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1}=? $$
How to find Laurent series of g(z) ?
$$ g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1} \hspace{10mm} \begin{cases} n \in N \\
0<a<1 \end{cases} $$
answer is :
$$ ...
6
votes
1answer
139 views
Power Series Expansion
I was flicking through a book on perturbation methods and saw a simple question asking the reader to expand the following expression for $f$ in a power series (up to the first 2 terms):
$f = (1 + ...
1
vote
1answer
45 views
What to plug in for n for this particular power series
I'm asked to find the power series representation of $f(x) = e ^ {-1/x^2}$ at $x = 0$.
getting the power series representation is easy of course.
$$\sum_{n = 0}^{\infty} \frac{(-1/x^2)^n}{ n!}$$
I ...
0
votes
2answers
395 views
Find the Taylor Series for $f(x)$ centered at a given value $a$
$$f(x) = \frac{6}{x}\,\, \mathrm{at}\,\, a = -4 .$$
Assume that $f$ has a power series expansion. Do not show that $R_n(x) -> 0$
I took the derivatives of f(x):
$$f(x) = 6/x$$
$$f'(x) = -6/x^2$$
...
1
vote
3answers
169 views
Power series representation of $x$?
This may not be a very good question, but I'm totally stumped.
I need to know the power series representation of $x$, or if there even is one.
I'll show you why:
I am trying to solve $y''+2xy'-y=x$ ...
4
votes
2answers
136 views
What's the background of this exercise?
I found this interesting exercise on a calculus book (Stewart)
Let
$$
u=1+\frac{x^3}{3!}+\frac{x^6}{6!}+\cdots
$$
$$
v=x+\frac{x^4}{4!}+\frac{x^7}{7!}+\cdots
$$
$$
...
1
vote
2answers
133 views
Use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $?
How do I use Residue Theorem to evaluate $
\
\oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz
\
$ where $C_3(0)$ is the circle of radius 3 centered at the origin, oriented in the counter-
clockwise ...
1
vote
1answer
204 views
Proof of the theorem about term-wise differentiation of power series
I have some doubts concerning the proof of the "term-wise differentiation of power series" theorem. Below, I first included 3 theorems that are used in the proof; then, I included the whole proof and ...
1
vote
0answers
284 views
Using power series to evaluate the integral of a piecewise function
I was self-studying Calculus, and the book I'm using asked me to solve the following integral using the technique of integration of power series:
$$\int_{0}^{0.25} g(x)dx \text{ where }
...
0
votes
3answers
130 views
Approximating cube root function for small values of $x$
How can one show that for small values of $x$, $\sqrt[3]{x+1}\approx1+\frac{x}{3}$?
1
vote
1answer
64 views
Do $\sum_{n=0}^{\infty}\frac{1}{a_{n}}$ and $\sum_{n=0}^{\infty}\frac{a_{n-1}}{a_{n}}$ and $\sum_{n=0}^{\infty}e^{-a_{n}}$MUST converge?
Let $\left\{a_{n}\right\}$ be a strictly increasing sequence of positive numbers,
Do $$\sum_{n=0}^{\infty}\frac{1}{a_{n}}$$
and
$$\sum_{n=0}^{\infty}\frac{a_{n-1}}{a_{n}}$$
and
...
16
votes
3answers
368 views
Math contest: Find number of roots of $F(x)=\frac{n}{2}$ involving a strange integral.
Edit summary: A good answer appeared. CW full answer added, based on given answers. Removing my ugly-looking attempts, as they still remain in the rev. history.
Here's a final-round calculus ...
0
votes
1answer
127 views
What's the limit of coefficient ratio for a reciprocating power series?
I have a question about the coefficient in the inverse of the power series.
Assume
$$
f=1-\sum_{i=1}^{\infty}(ck_i)x^i,
$$
where $c$ and $k_i$ are positive and $0<ck_i<1$ for any $i>0$. ...
1
vote
1answer
56 views
Any dominance between these two functions?
Let $f\left(x\right):=e^{x}+e^{-x}+2$ and $g_{\beta}\left(x\right):=4e^{\beta x^{2}}$.
Do there exist $a>0$ and $\beta>0$, such that $f\left(x\right)\le g\left(x\right)$
for all $x$, $0\le x\le ...
0
votes
1answer
200 views
Power series expansion
I recently had a problem. I know how to evaluate power series but I cannot seem to find an expansion for $\sqrt{x+1}$.
I've tried differentiating it, in order to bring it in reciprocal form but that ...
12
votes
2answers
298 views
Finding the convergence interval of $\sum\limits_{n=0}^{\infty} \frac{n!x^n}{n^n}$.
I want to find the convergence interval of the infinite series $\sum\limits_{n=0}^{\infty} \dfrac{n!x^n}{n^n}$.
I will use the ratio test: if I call $u_n = \dfrac{n!x^n}{n^n}$, the ratio test says ...
0
votes
2answers
230 views
The coefficient of a power series
Assume there has the power series $\sum_{n=1}^{\infty}a_nx^n$,the convergent radius of it $r>0$,prove:
If $a_1\neq 0$,and in a neighborhood of zero ...
