For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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0answers
36 views

What is the general form of a finite sequence?

I originally used: $a_0,a_1,a_2,...,a_n$ where $n$ is an integer greater than or equal to 0 but I've realized that this is actually an infinite sequence because n can go to $\infty$. How can I bound ...
1
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1answer
54 views

How to find a pattern in this recursive sequence algorithmically?

I'm trying to find the closed-form of a sequence algorithmically. Here is the recursive sequence: $$w_k=w_{k-2}+k, \forall k \in \Bbb{Z} | k \geq 3, w_1=1, w_2=2$$ which produces this sequence: ...
1
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1answer
62 views

Alternating cosine series, what is the closed form?

What is the closed form for this series: $$\sum _{n=0}^{\infty } \left(\frac{\cos \left(\frac{4 \pi}{2 n+1}\right)}{2 n+1}-\frac{\cos \left(\frac{4 \pi}{2 n+2}\right)}{2 n+2}\right)$$ if any? I am ...
1
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2answers
54 views

if one is bounded the other one is also bounded…

If $(a_n)$ is a bounded sequence of positive rational numbers and equivalent with sequence $ (b_n)$, show that $ (b_n)$ is also bounded. I was thinking that since $(a_n)$ and $ (b_n)$ are equivalent ...
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0answers
73 views

Test the uniform convergence of the series in indicated region

Test the uniform convergence of the series I tried to find $M_n$ such that $|\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}|\le M_n $ by using Abel's Theorem This is the question : Test the ...
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3answers
95 views

Finding the limit of a sequence by diagonalising a matrix

Consider the sequence described by: $\frac11 , \frac32 , \frac75 , ... ,\frac {a_{n}}{b_{n}}$ where $ a_{n+1} = a_n +2b_n $ and $b_{n+1} = a_n+b_n$ Find a matrix $A$ such that ...
2
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0answers
141 views

Is this sequence a Cauchy sequence?

Consider a continuous mapping $f: X \rightarrow X$, where $X \subset \mathbb{R}^n$ is compact and convex. Consider a sequence $\{x_k \in X\}_{k \geq 0}$ such that for all $k \geq 0$ and $h \geq 1$, ...
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3answers
102 views

Limit of the summand

I learnt that if $\displaystyle\lim\limits_{x\mathop\to\infty}f(x) \ne 0$ or if the limit does not exist then $\displaystyle\sum_{x\mathop=1}^{\infty}f(x)$ diverges. But suppose $f(x)$ takes the ...
2
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2answers
79 views

How to prove convergence of $a_n$ if $(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$?

Could you give me some hint how to conclude convergence of $a_n$ from this feature : $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ From $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ we may conclude that ...
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3answers
123 views

Taylor expansion for a multivariable function

\begin{align} T(x_1,\dots,x_d) &= \sum_{n_1=0}^\infty \sum_{n_2=0}^\infty \cdots \sum_{n_d = 0}^\infty \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + ...
0
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2answers
68 views

Multiplying and Dividing Series

For example, how do you compute the taylor series for $$e^x \sin x=\sum_{n=0}^{\infty} \frac {x^n}{n!} \sum_{n=0}^{\infty} (-1)^n\frac {x^{2n+1}}{(2n+1)!}$$ Of course I want the result to contain ...
1
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3answers
118 views

Simple series convergence/divergence

I have the following problem: $$\sum_{k=1}^{\infty}\frac{2^{k}k!}{k^{k}}$$ I only need to find whether the series converges or diverges. My initial thinking was to use the ratio test. I hit a stump ...
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5answers
225 views

infinitely descending natural numbers

Show that there is no infinitely descending sequence of natural numbers. I was thinking that there exists no infinite descending chain on the natural numbers, since every chain of natural numbers has ...
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1answer
29 views

Applying the derivative in the scope of Leibnitz test

Let, $$\sum_{n=1}a_{n}=\sum_{n=1} \frac{(-1)^n}{\ln(2n)}$$ By studing $\sum |a_{n}|$ I got divergence, so I couldt'n conclued anything about $\sum a_{n}$. Then I considered $b_{n}= \frac{1}{\ln(2n)}$ ...
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3answers
309 views

Use series to evaluate limit

limit as x approach infinity of $3(x^2)(e^{-2/x^2}-1)$. I don't know what series to use.
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4answers
43 views

Finding the Remainder

Given the polynomials $$P(x) = nx^n+(n-1)x^{n-1}+(n-2)x^{n-2}+\cdots+x+1$$ and $$Q(x)=x(x-1)^2$$ find the remainder of the division $\dfrac{P (x)}{Q (x)}$.
0
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2answers
84 views

What is the sum of this series given the closed form?

The closed form of a series I am trying to identify is: $$ a_n=\frac{250}{2n -1} $$ How could I get the sum of the series equation from this? I am used to geometric sequences and arithmetic sequences ...
1
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0answers
74 views

$\sum x_n$ is divergent , $ \lim x_n=0$ and the partial sums of $\sum x_n$ are bounded [duplicate]

Does there exist a divergent series $\sum x_n$ such that $ \lim x_n=0$ and the partial sums of $\sum x_n$ are bounded ?
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1answer
29 views

$\sum_{n=1}^{\infty}|a_{n} x^{n}| < \infty \implies \sum_{n=1}^{\infty} |a_{n}| (|x|^{n-1}+ |x^{n-2}y|+…+|y^{n-1}|) <\infty $?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$. Now, we let, $x, y \in \mathbb R$ with $x\neq ...
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3answers
50 views

$\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
0
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1answer
31 views

Finding the series solution for a second order ode

Find the series solution for $y''-2y'+2y=0$ Assuming that $y=\Sigma^{\infty}_{n=0} c_nx^n$I got the recurrence relation: $c_nn(n-1)-2c_{n-1}(n-1)+2c_{n-2}=0$ Therefore: $c_3=\frac13c_1-\frac23c_0$ ...
1
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1answer
39 views

Nature of the series $\sum _{n=1}^\infty (\sin n)^n $ , $\sum _{n=1}^\infty (\cos n)^n $

How do we discuss the convergence of the following series i)$\sum _{n=1}^\infty (\sin n)^n $ ii)$\sum _{n=1}^\infty (\cos n)^n $
1
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1answer
49 views

How to justify, $\sum_{n=1}^{\infty} a_{n} x^{n} - \sum_{n=1}^{\infty}a_{n}y^{n}=\sum_{n=1}^{\infty} a_{n} (x^{n}-y^{n})$?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$ and we let $K_{1}$ be a compact subset of ...
0
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0answers
29 views

Series $1 , 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$ [duplicate]

I've got this question here where I'm asked to find the limit of a sequence. But my sequence is : $$1 , 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$$ and so on... ...
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1answer
81 views

Sum of a Series (Calculus )

Evaluate the following sum as $n\to\infty$: $$a_n=\sum_{k=1}^n\frac{k(k+1)}{2x^{k-1}}, \quad|x|>1$$ Source: Exercise 14, http://www.mathem.pub.ro/_SITE_ELEVI/e-2005-a1.pdf . Thank you for ...
1
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1answer
45 views

Limit of sequence that is root of sums of powers

Suppose $z_1,\ldots,z_k$ are complex numbers with $|z_1|>\cdots>|z_k|$, and let $c_1,\ldots,c_k$ be non-zero complex numbers. Let $a_0,a_1,\ldots$ be the sequence defined by $$a_n=\sum_{i=1}^k ...
1
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3answers
78 views

Question on Uniform Conergence

I need to show that $\sum_{k=1}^\infty$$(\frac {x}{2})^k$ does not converge uniformly on (-2, 2) I know I have to show that $\sup_{x\in X}\lvert f_n(x)-f(x)\rvert\nrightarrow0 $ as ...
0
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1answer
62 views

Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
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2answers
62 views

Testing A Series For Convergence

Determine whether the series $\sum_{n=0}^{\infty} \frac{3n^2 + 2n + 1}{n^3 + 1}$ with n from 0 to infinity converges or diverges. So far I thought about dividing the numerator by the denominator, ...
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3answers
62 views

Is this a correct recursive sequence definition?

Take this definition: Is this definition of $s_k$ for $k\ge2$ correct? $s_k=6a_{k-1}-5a_{k-2}$ but where does the $a$ term come from? The book swaps $a$ and $s$ interchangeably.
3
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5answers
674 views

Why do some series converge and others diverge?

Why do some series converge and others diverge; what is the intuition behind this? For example, why does the harmonic series diverge, but the series concerning the Basel Problem converges? To ...
0
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1answer
25 views

help me finding the limit of sequence question

A 1= 1 and {A n}=root[1 + {A n-1}] B 1= 2 and {B n}=root[2*{B n-1}] help me Since I am studying math recently I need person`s help
0
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1answer
37 views

Construct a converging series that is substantially larger than a given converging series

This is more of a request for advice than a request for solution. Last night we were given the following and nobody figured it out in the time given (about 5 minutes). I think this is a problem many ...
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1answer
117 views

The general term formula $a_{n+1}=\dfrac{1+a_n^2}2$

Let $\ a_1=\dfrac 12\ $ and $\ a_{n+1}=\dfrac{a_n^2+1}2$, Could we find the general term formula $a_n$? If the anwer to the question $1$ is "NO", for $\left|a_n-\dfrac{n-1}{n+1}\right|$ and ...
4
votes
1answer
45 views

Formula for the following sum?

I simply wonder if it exists a formula for the sum $S_n = \sum_{k=1}^{n} k^k$ ? If it does, then what is it? If not, how do we know that?
3
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2answers
56 views

Is it true that, $|e^{x}-e^{y}|\leq C \cdot |x-y|$?

Define $f:\mathbb R \to \mathbb R$ such that $f(x)= e^{x}-1:= \sum_{n=1}^{\infty} \frac{x^{n}}{n!};$ for $x\in \mathbb R.$ My Question: Can we expect $|f(x)-f(y)|\leq |x-y| \cdot C;$ where $C$ is ...
2
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2answers
485 views

High Order Derivative Using Maclaurin Series

Use the Maclaurin series to solve the following: $$ \frac{d^6}{dx^6}(x^4e^{x^2}) $$ I got about halfway through the problem before getting stuck. I am not sure how to solve it... Any advice? Also, ...
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1answer
38 views

Problem Relating to Error in Series

For the following series, find the number of terms required to find the sum with error < 0.005, and find upper and lower bounds for the sum using a much smaller number of terms. ...
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4answers
70 views

Determine if the given sequence converges or diverges

Let $(x_n)$ be a sequence defined as $x_n = \frac{1}{n} \sum_{j=1}^{n} \frac{j+1}{j^2}$ . We want to know if $(x_n)$ converges. The trouble I am having here is that the sum depends on $n$. We know the ...
2
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3answers
110 views

Determine if $\sum\limits_{n=1}^{\infty}(1+\dfrac{2}{n})^n$ converges or diverges

I have an infinite series $\sum\limits_{n=1}^{\infty}(1+\frac{2}{n})^n$. I need to show if it converges or diverges using any test. I've tried applying all of the tests that I know, and hit dead ...
0
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1answer
43 views

Number of configurations in a constrained nested loops and configuration back from serial

Consider 4 counters looping the digits 0, 1, 2 to form the various "configurations", like in : ...
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1answer
39 views

How to prove that the space of finitely nonzero sequences of rational numbers is countable?

I was attempting to prove that $l^p$ is separable and concluded that since $c_{00}$ is dense in $l^p$ it should be easier to prove that $c_{00}$ is separable. Now since "$c_{00}$ with rational ...
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3answers
47 views

Comparison of two alternating series- Which is bigger

Imagine you have two finite alternating series. $$S_a=a_1-a_2+a_3-a_4+\cdots+a_n$$ $$S_b=b_1-b_2+b_3-b_4+\cdots+ b_n$$ Question: If $|a_i|>|b_i|$ is $S_a>S_b$?
2
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2answers
106 views

How could I have found the closed form of $\sum_{k=1}^n \frac{k}{(k+1)!}$ in advance?

If you calculate the first three sums, a pattern becomes clear revealing the closed form which is easily proven by induction: $$\sum_{k=1}^n \frac{k}{(k+1)!} = \frac{(n+1)!-1}{(n+1)!}$$ I’ve been ...
0
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4answers
78 views

How to reduce this series to a single equation?

Somehow, my textbook was about to reduce this series to a single equation: I know that you can use the equation $$S=\frac{n(n+1)}{2}$$ for the sum of the first n integers but I don't think this ...
1
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3answers
56 views

Need to know how: $L^{p}$ convergence for function.

I feel terrible at the moment as I have exhausted everything possible to understand this and I am still stuck. I have looked everywhere for some sort of resource to the concept at hand, and yet, they ...
0
votes
2answers
57 views

A question about using Squeeze Theorem to solve theoretical convergence question

Could you give me some hint how to deal with this question: Suppose $a_n\le b_n \le c_n$ for almost all n, $b_n\to L$, $c_n-a_n\to 0$. Prove: $a_n \to L,b_n \to L$. Well, if $a_n\to a, b_n \to b$ ...
0
votes
1answer
324 views

Summation of product of combinations

my question is, can the following series be solved $$ \sum_{i,j}^{} {a\choose i} {b \choose j} $$ where, (i+j) mod 3 =0 or i+j is multiple of 3 I need a generalized solution, i.e variables i,j,k... ...
0
votes
1answer
61 views

Is this construction already a contradiction

Let's say we have $\ell^p$ for $p>2$ and $\ell^2 \subsetneq \ell^p$. Let $U$ be a closed subspace of $\ell^2$. Is it possible that $U$ is isomorphic to $\ell^p$ as a Banach space?- I hardly think ...
3
votes
2answers
164 views

How to calculate $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac 1{4n}$?

Could you please help me calculate this limit: $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac 1{4n}$. My best try is : $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac ...