Tagged Questions
4
votes
1answer
39 views
trigonometric identity related to $ \sum_{n=1}^{\infty}\frac{\sin(n)\sin(n^{2})}{\sqrt{n}} $
As homework I was given the following series to check for convergence:
$ \displaystyle \sum_{n=1}^{\infty}\dfrac{\sin(n)\sin(n^{2})}{\sqrt{n}} $
and the tip was "use the appropriate identity".
...
5
votes
2answers
89 views
Calculate the limit of two interrelated sequences?
I'm given two sequences:
$$a_{n+1}=\frac{1+a_n+a_nb_n}{b_n},b_{n+1}=\frac{1+b_n+a_nb_n}{a_n}$$
as well as an initial condition $a_1=1$, $b_1=2$, and am told to find: $\displaystyle ...
1
vote
3answers
47 views
Which one is the correct series expansion?
Is
$$p^{n+1} = p^0+p^1+ \dots + p^n$$
or
$$p^{n+1} = p^0\times p^1\times \dots \times p^n\text{ ?}$$
I am confused.
please explain the correct one.
0
votes
2answers
70 views
Taylor series of $f(x)=\frac {e^x-1}{x}$
I am asked to expand $f(x)=\frac {e^x-1}{x}$ centered at 0 using the known Talyor series of functions.
How to simplify the function so that it can be expanded more easily?
3
votes
0answers
84 views
Compute limit of the sequence $x_n$
Let $(x_n)$ be a real sequence such that $x_0=a\in\mathbb{R},x_1=b\in\mathbb{R},x_{n+2}=-\dfrac{1}{2}\left(x_{n+1}-x_{n}^2\right)^2+x_{n}^4\;\forall n\in\mathbb{N} $ and $|x_n|\leq ...
13
votes
3answers
314 views
$\sum\limits_{\text{prime }p} 2^{-p}$ is an irrational number
I need help to prove the following result.
$\displaystyle\sum_{\text{prime }p} 2^{-p}$ is an irrational number.
0
votes
0answers
29 views
analysis: limit of product of sequences [duplicate]
I would really appreciate help with this question:
Show that if the limit $\lim_{n \to \infty} {a_n} = 0$ and sequence $\{b_n\}$ is bounded, then $\lim_{n\to \infty} a_nb_n =0$
thanks
2
votes
4answers
63 views
Approximation of alternating series $\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + …$
$\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + ...$
I am asked to find the no. of terms needed to approximate the partial sum to be within 0.0000001 from the convergent ...
0
votes
1answer
16 views
Geometric sequence, finding the first term using only the sum, the number of terms and value of one term.
In Geometric series: S = 56, a(2) = 16 and n = 3
S - sum, a(2) - second term, n - number of terms
Is it possible to get a(2) and a(3) from here? (If yes, hints would be awesome)
Thank You!
0
votes
2answers
30 views
Whether an infinite series can be tested by integral test
I am asked whether the following infinite series can be proved to be convergent by integral test.
$$\sum_{n=1}^\infty n e^{6 n}$$
so I integrate it
$$\int_1^{\infty}\ n e^{6n}\, dn$$
and find it ...
1
vote
2answers
39 views
Values of a parameter $x$ in an infinite series that makes it converge
I am required to find the values of $x$ in the following infinite series, which cause the series to converge.
$$\sum_{n=1}^\infty \frac{x^n}{\ln(n+1)}$$
I tried to use the ratio test, and found that ...
1
vote
1answer
44 views
Value of coefficients of the power series when radius of convergence is “less than 1” and “greater than or equal to 1”
Let $\sum_{n=1}^\infty c_n (x-a)^n $ be a power series. As "n" approaches infinity,the value of the coefficients "$ c_n $" may or may not be 0 when Radius of convergence R is such that 0< R ...
2
votes
2answers
41 views
power series and sequence
Let $\{ $a_n$ : n\geq 1\}$ be a sequence of real numbers such that radius of convergence of the power series : P(t) = $ \sum\limits_{n=0}^\infty a_n t^n $ satisfies $R > 0$.Then $ a_n \rightarrow ...
2
votes
2answers
31 views
Proving the coefficient of Power series is “0” always on given condition.
Suppose the power series $P(x) = \sum_{n=1}^\infty b_n x^n$ converges for $|x| \leq 1$ and that for some $c>0$ it is given that $$P(x)=0 \quad \forall x \;\text{such as}\;|x| < c$$ Show that ...
1
vote
2answers
48 views
is $\sum\limits _{n=1}^{\infty}\ln\left(\frac{\left(n+1\right)^{2}}{n\left(n+2\right)}\right)<\infty$?
I don't know why but I'm having a hard time determining whether this series
$$
\sum\limits _{n=1}^{\infty}\ln\left(\frac{\left(n+1\right)^{2}}{n\left(n+2\right)}\right)
$$
converges to a real limit.
...
3
votes
3answers
38 views
A problem on recurrence relation
Consider the sequence $$a_n = a_{n-1} a_{n-2} +n$$ for $n \geq 2$, with $a_0 = 1$ and $a_1 = 1$. Is $a_{2011}$ odd?
By writing all the terms of the sequence I see that $a_n$ is odd when $n$ is odd ...
1
vote
4answers
60 views
Sequence Diverging: Where's My Mistake?
EDIT: Question answered; I misread my own handwriting when copying my notes into LaTeX. Thanks!
I'm trying to show that $a_{n}$ diverges. The equation I arrive at does not diverge. Where did I go ...
2
votes
1answer
30 views
how do I show $\frac{1}{\sum_{\tau=1}^t\lambda^{t-\tau}} = \frac{1-\lambda}{1-\lambda^t}$
How can I show that $\frac{1}{\sum_{\tau=1}^t\lambda^{t-\tau}} = \frac{1-\lambda}{1-\lambda^t}$ ?
Any help on how to get started will be appreciated
A very long derivation just states that this is ...
13
votes
3answers
219 views
Proving a trig infinite sum using integration
How can I prove the following using integration and elementary functions?
Prove that:
$$\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{n} = \frac{\pi}{2} - \frac{\theta}{2}$$
$0 < \theta < 2\pi$
2
votes
3answers
44 views
prove that : $ \sum_{n=0}^\infty |x_n|^2 = +\infty \Rightarrow \sum_{n=0}^\infty |x_n| = +\infty $
i wanted to prove initially that a function is well defined and i concluded that it's enough to prove this statement for $x_n$ a sequence :
$ \sum_{n=0}^\infty |x_n|^2 = +\infty \Rightarrow ...
0
votes
1answer
38 views
Calculating a sum of series. [duplicate]
I have the following series:
$\sum_{n=1}^{\infty} \left(1\over2^{n-1}\right) $
I have to calculate its sum. I don't know how to do so. I'd like to get helped. thanks in advance.
3
votes
1answer
53 views
Limits with sums and integrals
It's one of my homework exercises that is rather problematic to me. Apparently the last thing to do is to squeeze it but I don't see yet how to do that. Could you help?
...
0
votes
1answer
42 views
For what values of p the series is convergent.
The series is:
$$\sum_{n=2}^{\infty} \dfrac{1}{(n\log(n))^p}$$
I use Cauchy's condensation criterion but this give the next series:
$$\sum_{n=2}^{\infty} \dfrac{1}{(n\log(2))^p}$$
$\log(2)^p$ is a ...
0
votes
2answers
53 views
Geometric series proof?
How do I show that this given series is geometric:
$$(1/2) + (1/2)^4 + (1/2)^7 + (1/2)^{10} + \cdots$$
and therefore calculate the sum to infinity
8
votes
1answer
163 views
Function mapping challange
For a given set $A=\{1, 2, 3, 4, \ldots, n\}$, find the number of non-constant
mappings (associations ) $f$ from $A$ to $A$ such that $f(k) \leq f(k + 1)$
and $f(k) = f(f(k + 1))$.
This is ...
1
vote
1answer
26 views
On the limit of a Minkowski sum
Consider an open set $\mathcal{O} \subseteq \mathbb{R}^n$.
I am wondering if the set $$ \mathcal{S} := \lim_{k \rightarrow \infty} \ \mathcal{O} + \frac{1}{k} \mathbb{B} $$
is open or closed.
With ...
2
votes
1answer
69 views
Computation of standard series
I am stuck on the computation of the following sum:
$\sum_{k=1}^\infty e^{-n^2}\cos(n)$. Simple tricks fail and also i have no idea how to fit it for Fourier series. Are there any other ways?
1
vote
2answers
42 views
Prove that the series is non-absolutely convergent.
$$a_n = \int_{(2n-2)\pi}^{(2n-1)\pi} \dfrac{\sin t}{t} dt$$
The series is $$\sum_{n=1}^{\infty}a_n$$
I tried using the Cauchy criterion, and this let me with the next inequality:
$$\left| S_m - S_n ...
0
votes
1answer
64 views
For which $x$ values this series $\sum_{n=1}^{\infty}\frac 1n \cos^2(nx)$ is convergent.
I have tried using Dirichlet's Test with $\cos^2nx = \dfrac{1+\cos(2nx)}{2} = \frac 12 + \dfrac{\cos(2nx)}{2}$, this show that I can't bound the partial sums.
What another test may I apply?
1
vote
2answers
57 views
ratio test and divergence
I need to be able to prove that this series converges. I know I need to use the ratio test but I do not know how to go about doing it.
Any help is much appreciated! thank you
3
votes
1answer
41 views
Struggling to understand a couple of concepts with series
I have two questions: neither of which are homework problems but certainly pertain to my ability to do the homework. The first regards the harmonic series. The question has been answered often here ...
2
votes
2answers
84 views
Using the Banach Fixed Point Theorem to prove convergence of a sequence
Use the Banach fixed point theorem to show that
the following sequence converges. What is the limit of this
sequence?
$$\left(\frac{1}{3},
\frac{1}{3+\frac{1}{3}},
...
1
vote
4answers
88 views
Finding generating function for the recurrence $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n - 1}$
I am trying to find generating function for the recurrence:
$a_0 = 1$,
$a_n = {n \choose 2} + 3a_{n - 1}$ for every $n \ge 1$.
It looks like this:
$a_0 = 1$
$a_1 = {1 \choose 2} + 3$
$a_2 = {2 ...
0
votes
0answers
35 views
Identifying the term in a sequence [closed]
I need help with this assignment of Identifying the next term in the sequence below:
12, 15, -1, 17, 600, 38,...?
0
votes
1answer
29 views
Maclauren Series and taylor polynomials
Question:
Suppose that the function $k(x)$ has a maclauren series that converges $\left(-\frac{1}{2} , \frac{1}{2}\right]$ and you are told that $|k^{(n)}(x)| \leq 10$ at all $|x| \leq ...
0
votes
1answer
48 views
How to prove a telescoping series converges ???
Let ${a_k}$ be a real sequence, such that $a_k \rightarrow 3, $as k$ $$\rightarrow \infty$ .
Prove that $\sum_{k=1}^\infty (a_k- a_{k+3}) = a_1 + a_2 +a_3 - 9$.
I know this is a telescoping series ...
1
vote
0answers
76 views
Why does Σ1/x diverge? [duplicate]
Why does the following series diverge?
$$\sum_{n = 1}^\infty\frac{1}{n}$$
I've tried to make sense of it, but can't seem to wrap my head around it.
Thanks.
0
votes
1answer
13 views
Upper Bounds of Two Interdependant Recursive Sequences
For a pair of real numbers $\alpha$ and $\beta$, I need to prove that for the sequences
$a_n = (-\alpha)a_{n-1} +b_{n-1}$
$b_n = (-\beta)a_{n-1}$
an upper bound exists with a form similar ...
3
votes
4answers
177 views
Check whether a infinite series is convergent or diverge?
I have the following infinite series:
$$\sum_{n=1}^{\infty}{\left(1-\frac{1}{2n}\right)^{n^2}}.$$
How can I check whether this infinite series is convergent or diverging?
0
votes
2answers
59 views
Show that cosh(2) is between two values.
I'm reviewing for exams and this question has got me stumped:
Show that:
$3\dfrac{2}{3} \leq \cosh(2) \leq 3\dfrac{2}{3} + 0.1$
I've determined the series form of cosh(x) to be:
...
1
vote
2answers
58 views
Question about a infinite series
Could you please explain how come:
$$\sum_{i=1}^\infty(0.5)^{i+1}(i+1)=2$$
How come the answer is not inf? And why 2?
Any help will be appreciated.
1
vote
2answers
49 views
How to think about solving recurrences?
I am having trouble finding a closed-form solution to the following recurrence for $T(i)$, $0\le i\le n$.
$$T(0) = T(1) + 2,\quad T(n) = 0$$
and
$$T(i) = {T(i+1)\over 2} + {T(i-1)\over 2} + 1,\quad ...
1
vote
2answers
30 views
Convergence of infinite series with p-test and constant
My question is: Does the infinite series
$\sum_{n=1}^\infty \frac{1}{n^{\frac{4}{5}}+10^{10}}$
converge or diverge?
I know that $\frac{1}{n^{\frac{4}{5}}}$ diverges by the $p$-test, and that adding ...
2
votes
2answers
56 views
Finding formula for the nth partial sum
A few days ago, I asked for some clarification about pattern recognition and the n-th partial sum for infinite series. Although the explanation given was top-notch (thanks again), I'm still having ...
1
vote
1answer
69 views
Summation of Arithmetic-Geometric Series
I've been working through my homework paper, and I've come across this question. Now I'm confident in what I have done for the most part, but I am stuck at the end.
I have this recurrence relation, ...
2
votes
2answers
71 views
Real analysis - converging sequence [duplicate]
My answer
Solution
1).
$Let\; \epsilon = L/2 > 0 \mbox{thus by definition of}\; x_m→L, \mbox{there exists}\; a \;n_o∈ N \;\mbox{such that }∀m>n_o\; \\
|x_m - L|< ε\\
-ε <|x_m - L| < ...
2
votes
3answers
120 views
Calculating the Fourier series of $x^{3}$
I was given as homework to calculate the Fourier series of $x^{3}$.
I know, in general, how to obtain the coefficients of the series using
integration with $$\sin(nx),\cos(nx)$$ multiplied by the ...
3
votes
3answers
98 views
Finding a limit where binomial coefficients appear as powers.
This one is for my mate that is 2 years older than me. Could you help please?
$$\lim_{n\to\infty}\frac{\sqrt[\uproot{3}\Large 2^n]{n^{\textstyle\binom{n}{0}}\cdot ...
0
votes
2answers
110 views
How to calculate: $\displaystyle \frac{1}{2}+\frac{3}{4}+\frac{5}{6}+…+\frac{99}{100} $
How can I calculate value of $\displaystyle \frac{1}{2}+\frac{3}{4}+\frac{5}{6}+.....+\frac{97}{98}+\frac{99}{100}$.
My try:: We Can write it as $\displaystyle \sum_{r=1}^{100}\frac{2r-1}{2r} = ...
1
vote
1answer
40 views
Understanding the fundamentals of pattern recognition
I'm learning now about sequences and series: patterns in short. This is part of my Calc II class. I'm finding I'm having difficulty in detecting all of the patterns that my text book is asking me to ...
