For questions about recurrence relations, convergence tests, and identifying sequences

learn more… | top users | synonyms (3)

0
votes
0answers
4 views

Determine whether following series converges::

My thoughts: When n approaches infinity so given series diverges by Comparison test because $\frac{1}{n}$ diverges. Firstly, I wonder if I am correct on this one. And would there be another, ...
1
vote
0answers
10 views

On the possible values of $\sum\varepsilon_na_n$, where $\varepsilon_n=\pm1$ (i.e., changing signs of the original series)

I have used the following result in an answer on this site. Suppose that $a_n>0$ are positive real numbers such that $\sum\limits_{n=1}^\infty a_n=+\infty$ and $\lim\limits_{n\to\infty} ...
3
votes
3answers
24 views

Determine whether following series converges:

$$ \sum_{n=1}^\infty \frac{\sqrt{n}}{\sqrt{n^3}-i} $$ I determined that series diverges, because it's less than $\frac{1}{n}$ (I assumed that $i$ has no influence here) and $\frac{1}{n}$ diverges, ...
0
votes
0answers
4 views

understanding of discrete prolate spheroidal sequences

i would like to know some details about discrete prolate spheroidal sequences or Slepian sequences,because they have application in multitaper method used in DSP,as i undersood they are time varying ...
2
votes
2answers
26 views

Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?

If I have a sequence {$a_n$} that has the property of $\lim(a_{n+1}-a_n)=0$, does that make it a Cauchy Sequence. I think it doesn't because $a_n = \sum_{k=1}^n \frac{1}{k}$ is a counter example. ...
2
votes
2answers
36 views

$a_1=3$ and $a_{n+1}=\frac{a_n}{2} + \frac{1}{a_n}$. Show that it monotonically decreases and find the limit.

What I've done so far: I have proved that this sequence is bounded below by 0, which is a very rough estimate. I know that the infimum is $\sqrt2$. Anyway, the question first asks me to prove that ...
1
vote
0answers
24 views

Proving a series is greater than zero

I wish to prove that an equation is greater than zero. Let ...
0
votes
0answers
16 views

Relation between coefficients of Maclaurin series and recurrence relations

Suppose $I_{2n}=\int_0^{\pi}x\sin^{2n}x \;dx$. Then one can obtain the following: $$I_0= \frac{\pi ^2}{2}, I_1= \frac{\pi ^2}{4}, I_2= \frac{3\pi ^2}{16}, I_3= \frac{5\pi ^2}{32},\ldots, I_n= ...
0
votes
5answers
33 views

How to create alternating series with happening every two terms

I'm looking for a technique for creating alternating negatives and positives in a series. Specifically: when n=1, the answer is +, n=2 is +, n=3 is -, n=4 is -... etc. I have every other part of the ...
0
votes
2answers
33 views

How to simplify the expression with sigma notation?

$$\sum\limits_{i=1}^n (n-i+1)(2i-1)^2= \frac{n(2n^3+4n^2+n-1)}{6}$$ how does this work? Could anybody show the details.
1
vote
4answers
38 views

$s_1 = 1$ and $s_{n+1}=(\frac{n}{n+1})s_n^2$ monotonically decreases?

Hi I came across this question on page 65 of Elementary Analysis by Kenneth A.Ross. I am given that $s_1 = 1$ and I need to show that $s_{n+1} = (\frac{n}{n+1}) s_n^2$ monotonically decreases. I'm ...
2
votes
2answers
33 views

If $\sum_{n=1}^\infty a_n$ converges and $\sum_{n=1}^\infty \frac{\sqrt a_n}{n^p}$ diverges, then p $\in$ {?}

Let {$a_n$} be a sequence of non-negative real numbers such that the series $$\sum_{n=1}^\infty a_n$$ is convergent. If p is a real number such that the series $$\sum_{n=1}^\infty \frac{\sqrt ...
0
votes
3answers
67 views

How to prove $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ converges?

Could you please give me some hint how to prove this statement: If $0<a_n<1$ for each n, then $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ ...
0
votes
0answers
21 views

The function $\sum_{i=1}^\infty i \exp(-a i^2)$

Consider the function $\phi(a)=\sum_{i=1}^\infty i \exp(-a i^2)$ with $a>0$ (see below for the physical motivation). I was very surprised that Mathematica didn't have an expression for a function ...
0
votes
0answers
18 views

Binomial series for $2^{n-1}\cos^n\vartheta$ and $2^{n-1}(-1)^{\frac{n}{2}}\sin^n\vartheta$

Can somebody confirm for me whether the following series are correct? $$2^{n-1}\cos^n\vartheta=\cos ...
1
vote
1answer
16 views

Interest Accumulation - Geometric Sequence

Hello I have just worked a question in which I get an answer different to the answer in my book. The question states: If a person deposits 500 at the end of each month for 20 years at an AER of ...
1
vote
1answer
24 views

Find the limit of $(a_k)_k$ where $a_k = \lim_{n\to\infty} \frac1n\sum_{m=1}^{kn} \mathrm{exp}(\frac12.\frac{m^2}{n^2})$

For $k \ge 1$, let $$a_k = \lim_{n\to\infty} \frac1n\sum_{m=1}^{kn} e^{\frac12.\frac{m^2}{n^2}}$$ Find the value of $$\lim_{k\to\infty} a_k$$
0
votes
1answer
27 views

Inverse of a lower triangular Toeplitz matrix vs. the matrix size

I am recently trying to find the inverse of the lower triangular Toeplitz matrix ($\mathbf{A}$), with some special elements: $$ \mathbf{A}_{M\times M}=\left[\begin{array}{ccccc} 1\\ -a_{1} & 1\\ ...
0
votes
0answers
20 views

how to find if a number is closer to a series of numbers?

Lets say you have a random series of real numbers. How to determine if a random test sample is closer to that random series within an acceptable threshold?
2
votes
1answer
28 views

The “trick” in the Herglotz trick

In How does the Herglotz trick work?, is explained as in "Proofs from THE BOOK" by Aigner and Ziegler, but the "trick" itself I found to be not so clear. The trick says: It follows from (4) ...
1
vote
0answers
34 views

I am having a difficult time trying to generate this. [on hold]

https://oeis.org/A008004 Is there a formula for this sequence? The links in the page have brought me to this: ...
1
vote
2answers
63 views

a question about limit, I am struggling with this!

Suppose that {$a_n$}is a sequence of positive numbers.For each n which is a natural number,let $b_n$=($a_1+a_2+......a_n$)/n,prove that $\sum b_n$ diverges to $+\infty$. This question is my homework, ...
1
vote
1answer
45 views

Value of divergent series?

Let $\{a_n\}$ be a positive, convergent sequence. We consider the sequence of partial sum $\{s_n\}: s_n = \sum_{k=1}^n a_n$. Clearly $\{s_n\}$ is strictly increasing and therefore $\sum_{n=1}^\infty ...
0
votes
1answer
33 views

Absolute convergence does not implies convergence

Find a space and a series that converges absolutely but it does not converges. It is clear that the space can't complete or Banach.
3
votes
1answer
45 views

Sum as an integral

Recently I have encountered weird notation that I don't see into. When I have some infinite sum $$\sum_{n=1}^{\infty}f(n)$$ I would rewrite it without thinking to the integral form like this ...
1
vote
0answers
28 views

Prove that the limit of the function a sequence is the same as the function of the limit of the sequence [duplicate]

Assume that $\lim_{n\to \infty}a_n=a.$ Suppose that the function $f$ is continuous everywhere including at $a$. Form the sequence $(f(a_n))_{n=1}^{\infty}$. Prove that $\lim_{n\to ...
1
vote
4answers
114 views

Does the sequence $\cfrac{n!}{\pi^n}$ converge or diverge and why? [on hold]

The problem states: Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges. $$C_n = \frac{n!}{\pi^n}$$
2
votes
4answers
19 views

Need help finding the limit of geometric serires

I'm learning some series tests in calculus and I can't completely figure this out. I know it's easier than i'm making it. Here's the question: Determine whether the geometric series is divergent ...
0
votes
2answers
21 views

Summation with non integer induces

All of the sums I've encountered so far have been functions $f:Z\rightarrow$ (any other set). Or in other words the sums are in the form of $\sum_{j\in Z\lor j\in N}^n a_j=a_0+a_1+...+a_n$. This ...
1
vote
2answers
33 views

Is the $\sum_{n=1}^{\infty} \frac{(2n+1)^{1/2}}{n^2}$ convergent or divergent?

For this question I am not really sure which test to use to determine this. I was thinking the comparison or limit comparison test but it doesn't seem to be working. I was wondering what the steps are ...
2
votes
3answers
61 views

How to show a sequence converges

Let $u_n$ be a bounded sequence of real numbers. Suppose that $$\lim_{n \to \infty} u_n + \frac{u_{2n}}{2} = 1$$ Show that $u_n$ converges. Can someone provide some hints or insight to this ...
0
votes
0answers
13 views

Change of order of limit and function

Let $\Omega\subset\mathbb{R}^n$ be a open and bounded domain. Suppose that $f(x)$ is a $C^1$ function for $x\in\Omega$ and $\{ x_k \}_{k=1}^{k=\infty}\in\Omega$ is a sequence with ...
0
votes
1answer
25 views

Equality of two expressions

Is the following true for even integer n = 2m > 1? $\sum_{k=1}^m 2^{2k-1} \left( {\begin{array}{*{20}c} n \\ 2k-1 \\ \end{array}} \right) = \sum_{k=1}^m 2^{2k} \left( {\begin{array}{*{20}c} n ...
0
votes
1answer
44 views

Algebraic mean problem

The Question is: $27pqr \geq (p+q+r)^3$ and $3p+4q+5r=12$, then what is the greatest value of $p^3+q^4+r^5$? How do i solve this problem? Im think harmonic mean has to be used along with geometric ...
0
votes
2answers
59 views

Proof of $(1-e^{ix})^{-1}$

In G.H. Hardy's book 'Divergent Series' there is a claim that $(1-e^{ix})^{-1} = \frac {1}{2} + \frac {1}{2} i \cot (\frac {1} {2} x) $ I, for the life of me, can't get past showing that ...
2
votes
1answer
31 views

Some special tests for convergence.

Please, I need some explanations on some special tests for convergence of a series. For example, the ratio test, comparison test and root test. Ratio test: let the limit as $n$ tends to $\infty$ of ...
1
vote
2answers
37 views

Sum of this eries: $\sum_{k=1}^{\infty}kp(1-p)^{k-1}$

$\sum_{k=1}^{\infty}kp(1-p)^{k-1}$ Can someone help me evaluate this sum? I couldn't even start, I have just written down the first couple of elements, but didn't help either. Thanks!
2
votes
1answer
27 views

Convergent series multiplied by $n$

this one should be really simple... Let $\mathbb a_n$ be a sequence of strictly positive real numbers such that $\sum_{n\in\mathbb N}a_n$ is finite, i.e., the limit $\lim_{k\to\infty}\sum_{n=1}^ka_n$ ...
0
votes
0answers
18 views

ratio test for convergence of series, different versions

In lecture we had the ratio test: Let $(a_k)$ be a sequence in $\mathbb{K} \in\{ \mathbb{R}, \mathbb{C}\}, a_k\not= 0$ for all $k \ge k_0$, where $k_0\in \mathbb{N}$. (I) If there is a $q\in (0,1)$, ...
0
votes
1answer
17 views

Multiplication-Shift cipher Decyrpt

I am given sets of numbers to decrypt into letters, 446,882,915 ... these are the first 3. So I've been given K=2, a = 68 and shift b = 7. I've been given alphabet of n = 35 symbols. I know the ...
3
votes
1answer
78 views

$\lim_{x \to \infty} \frac{3+ 2 \sin (x)}{4+5x \sin(x)} $

A friend asked me to compute the following limit: Problem: Compute: $$ \lim_{x \to \infty} \frac{3+ 2 \sin(x)}{4+5 x \sin (x)}$$ I was clueless at first, so I gave it some thought and considered ...
1
vote
0answers
39 views

Turning a summation into an integral

I have a summation of the form: $$y(x) = \sum\limits_{h=-L}^L\frac{A(h)\cdot R(h)^2}{((x-h)^2+R(h)^2)^{3/2}}$$ Where I wish to solve/optimise $R(h)$ (leaving $A(h) = const/h$) or $R(h)$ and $A(h)$ ...
0
votes
0answers
37 views

Does the geometric sum formula have any useful variants?

Suppose $a$ is a constant sequence in $\mathbb{C}$ with constant value $A \in \mathbb{C}$. Then the geometric sum formula says: for all natural $n$ and all complex $z$, we have $$\sum_{k = ...
1
vote
1answer
69 views

Convergence of $\sum^\infty_{n=1}\frac{a_n}{1+a_n^2}$

We have a positive series $\displaystyle\sum^\infty_{n=1}a_n$. is the following series converge or diverge ?$$\displaystyle\sum^\infty_{n=1}\frac{a_n}{1+a_n^2}$$ ...
2
votes
3answers
36 views

Expression generating $\left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + \frac{333}{1000}, \dots \right)$

I'm looking for a closed-form expression (in terms of $n$), that will give the sequence $$ (s_n) = \left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + ...
3
votes
2answers
43 views

Convergence of $\sum^\infty_{n=1}\frac{a_n}{1+n^2a_n}$

We have a positive series $\displaystyle\sum^\infty_{n=1}a_n$. is the following series converge or diverge ?$$\displaystyle\sum^\infty_{n=1}\frac{a_n}{1+n^2a_n}$$ Suppose ...
3
votes
4answers
42 views

Showing limit of sequence $\left(\frac{3}{10}, \frac{33}{100}, \frac{333}{1000}, \dots\right)$

I'm trying to calculate the limit of the following sequence: $$ (s_n) = \left(\frac{3}{10}, \frac{33}{100}, \frac{333}{1000}, \dots\right). $$ Clearly, $(s_n) \to 1/3$, but I'm not sure how to show ...
1
vote
2answers
32 views

Convergence of the complex sequence $c_{n}=\left ( \frac{1}{\sqrt{2}}(1+i) \right )^{n}$?

I got an exercise to determine the sequence converges or not, $c_{n}=\left ( \frac{1}{\sqrt{2}}(1+i) \right )^{n}\in \mathbb{C}$. I re-wrote it to $$c_{n}=\left |c_{n} \right ...
0
votes
3answers
27 views

find the sum of series of $\sum_{k=0}^{\infty}\frac{4^k-3^k}{5^k}$ [on hold]

Find the sum of series $\sum_{k=0}^{\infty}\frac{4^k-3^k}{5^k}$
3
votes
1answer
113 views

How was this sequence discovered?

Let $N$ be a positive integer and consider the following rational sequence for $n \ge 0$: $$ a_{n+1} = \frac{N a_n + N}{a_n + N}, a_0 \in \Bbb{Q}. $$ If $-\sqrt{N} < a_0 < \sqrt{N}$, then ...