1
vote
6answers
211 views

How to prove a sequence does not converge?

I want to show that the sequence $$a_n=\frac{1}{n}+(-1)^n$$ does not converge to a limit. I know that if a sequence $\left( a_n\right)_{n \in \mathbb{N}}$ converges to a limit L, then ...
0
votes
1answer
30 views

Prove that all subsequential limits are contained within a closed interval

Let $a, b$ be two real numbers such that $a < b$, and suppose that $(s_n)_{n=1}^\infty$ is a sequence such that $\forall\,\, n\,\, a \leq s_n \leq b$. Prove that all subsequential limits are ...
2
votes
1answer
29 views

Verification of identity $2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$

Is this identity true? $$2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$$ If so, how to prove it? Could you provide me a ...
8
votes
2answers
109 views

Prove that $\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$

How to prove the following identity $$\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$$ I am totally clueless in this one. Would ...
-4
votes
2answers
40 views

Is there an easy way to check convergence of real sequence? [closed]

How to check convergence of a sequence (for example, of $x_n =[nx]/n$ for a fixed $x \in \mathbb R$.) For series we can use various tests to check its convergence which are not available for ...
1
vote
3answers
74 views

Proof that a sequence is convergent

I'm asked to prove the convergence of the sequence $$X_n=\left(1+\frac12\right)\left(1+\frac14\right)\left(1+\frac18\right)\cdots\left(1+\frac{1}{2^n}\right)$$ I proved that it is increasing through ...
1
vote
1answer
35 views

Convergence of $\sum_{n=0}^\infty (-1)^n (e-(1+\frac{1}{n})^n)$

Does $\sum_{n=0}^\infty (-1)^n (e-(1+\frac{1}{n})^n)$ converge absolutely, conditionally, or diverge? Attempt: Yes, by the ratio test we have $$ \lim_{n \to \infty} \left| \frac{(-1)^{n+1} ...
1
vote
1answer
39 views

Proving a Special Case of a Limit Theorem

I'm having trouble proving a special case of the limit theorem below. I attempted a proof by contradiction that appears to me to make sense in the first direction but I'm not able to come up with ...
3
votes
4answers
161 views

Show a sequence such that $\lim_{\ N \to \infty} \sum_{n=1}^{N} \lvert a_n-a_{n+1}\rvert< \infty$, is Cauchy

Attempt. Rewriting this we have, $$\sum_{n=1}^{\infty} \lvert a_n-a_{n+1}\rvert< \infty \,\,\,\Longrightarrow\,\,\, \exists N \in \mathbb{N}\ \ s.t,\ \ \sum_{n \geq N}^{\infty} \lvert ...
1
vote
0answers
81 views

Prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent.

I need to prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent, and find its limit using this theorem: Let $f:E \to R$ with $x_{0} \in E$ an accumulation point of $E$. Then these are equivalent: 1)$f$ ...
0
votes
1answer
37 views

Proving a sequence is increasing and converging as $n\to \infty$.

Suppose that $x_0 \in (-1,0)$ and $x_n=\sqrt{x_{n-1}+1}-1$ for $n \in \mathbb N$. Prove that $x_n \uparrow 0$ as $n\to \infty$. What happens when $x_0 \in [-1,0]$? Before this, the problems I did had ...
0
votes
3answers
26 views

Bounded Sequences and Extrapolation of Convergence From Related Sequences

I'm considering some sequence $S_n$ which is bounded, and I want to prove that $S_n/n$ is convergent. I'm thinking that I could simply take $lim_{n \to \infty} S_n/n$ and simplify this to $(lim_{n \to ...
0
votes
1answer
19 views

Proof limit of ratio of sequence .

Prove that as $n\to\infty$ $$\frac{1}{x_n} \to \frac{1}{x}$$ where we are also given $x_n \to x$, and $x_n,x\neq0$ Attempt: Suppose $x_n \to x$. Then for every $\epsilon > 0$, there exists a ...
0
votes
1answer
25 views

Square root of Sequence approaches square root value.

Suppose that $x$ is a real number, and $x_n\geq 0$, and $x_n→x$ as n grows. Prove that $\sqrt {x_n}→\sqrt x$ as $n$ grows. Attempt: Case 1: $x = 0$. Suppose that $x$ is a real number, and $x_n \geq ...
2
votes
1answer
49 views

Proving $\sum_{k=1}^{\infty}\frac{\sin kx}{x}=\frac{\pi-x}{2}$ for $0\le x\le 2\pi$

Refer to this OP: Sign of a series, we have the following equation \begin{equation} \sum_{k=1}^{\infty}\frac{\sin kx}{k}=\frac{\pi-x}{2} \end{equation} defined for $0\le x\le 2\pi$. Here is ...
0
votes
3answers
18 views

Sequences and Series - AP and GP

Question: If a,b,c are in GP and $$a^{1/x} = b^{1/y} = c^{1/z}$$ prove that x,y,z are in AP I tried writing b and c in terms of a, by assuming a common ratio r, however, I was unable to proceed from ...
0
votes
1answer
292 views

provide a combinatorial proof that $C_{n+1} = C_0C_n + C_1C_{n-1} + …. + C_kC_{n-k} + …C_nC_0$

(a) Let $C_n$ denote the number of ways of writing a valid list of open and closed parentheses of length $2n$ (valid means that at any point along the list, the number of open parentheses must be ...
0
votes
1answer
59 views

How to show that $\lim_{n \to \infty} \frac{\binom{n+k}{k}}{(n+k)^k} = \frac{1}{k!}$?

Given that $k\in\mathbb{N}$, my question is how to prove that this sequence converge to $\frac{1}{k!}$: $$\left\{ \frac{n+k \choose k}{(n+k)^{k}} \right\}_{n\in\mathbb{N}}.$$ I have this attempt:
1
vote
2answers
55 views

Sequences and series

If $p, q, r$ are in G.P. and the equations: $$px^2 + 2qx + r = 0$$ $$dx^2 + 2ex + f = 0$$ Have a common root, then show that $$\frac{d}{p}, \frac{e}{q}, \frac{f}r$$ are in A.P. Well I tried taking ...
2
votes
1answer
37 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
1
vote
7answers
121 views

Error in proving of the formula the sum of squares

Given formula $$ \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6} $$ And I tried to prove it in that way: $$ \sum_{k=1}^n (k^2)'=2\sum_{k=1}^n k=2(\frac{n(n+1)}{2})=n^2+n $$ $$ \int (n^2+n)\ \text d ...
3
votes
3answers
140 views

Convergent or divergent $\sum_{n=1}^{\infty} \frac{e^nn!}{n^n}$?

Any suggestion/hint, not the whole solution, how to determine convergence/divergence of $$ \sum_{n=1}^{\infty}\dfrac{e^n \cdot n!}{n^n} $$ I'm currently stuck.
1
vote
2answers
59 views

How to prove that the sequence is decreasing $a_{n}=\frac{ln(n)}{n^2}$

Is my way/proof good and completely mathematically rigorous? $a_{n}=\frac{ln(n)}{n^2}$ --> $a_{n+1}=\frac{ln(n+1)}{(n+1)^2}$ $\frac{ln(n)}{n^2} > \frac{ln(n+1)}{(n+1)^2}$ ...
2
votes
4answers
466 views

Proving 7n+5 is never a cubic number?

This is from a question that starts with: An arithmetic progression of integers an is one in which $a_n=a_0+nd$, where $a_0$ and $d$ are integers and n takes successive values $0, 1, 2, \cdots$ Prove ...
1
vote
1answer
35 views

Arithmetic progression and proofs

Here is the question I am stuck on An arithmetic progression of integers an is one in which $a_n = a_0 + nd$, where a_0 and d are integers and n takes successive values 0, 1, 2.... Proof that if one ...
19
votes
3answers
352 views

How to prove $ \lim_{n \to \infty} e^n \cdot \left( \sum_{k=0}^{n-1} ({k-n \over e})^k/k! \right)- 2 \cdot n = \frac 23$?

I observed for the function $$ f(n)= e^n \sum_{k=0}^{n-1}\left(\dfrac{k - n}{e}\right)^k \cdot \dfrac{1}{k!} \tag 1$$ with small $n$ that ...
0
votes
3answers
66 views

Convergence of sequence method, Math behind intuition

Now I want to find convergence of a sequence: $$ \lim_{n \to \infty} \sqrt[n]{4^n + 5^n}$$ Now I am pretty sure I have solved this using logic on inspection: $4^n \ll 5^n$ as $n\rightarrow\infty$, ...
3
votes
1answer
73 views

Epsilon-delta proof for a sum

Consider $\displaystyle \begin{array}{ccccc} f & : & \mathbb R^+ & \to & \mathbb R \\ & & x & \mapsto & \frac{x}{\sqrt{1+x}} \\ \end{array}$ Prove that ...
3
votes
2answers
78 views

Passing a derivative through a limit.

After searching around on the net and on SE I have not found a satisfactory answer. Let $f_n: D \to \mathbb R$ be a sequence of functions. What assumptions, aside from $f$ being differentiable, do we ...
0
votes
2answers
38 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
1
vote
2answers
24 views

Series, limits and convergence.

Theorem $\,\bf3.3.1.\;$ If the series $$\sum_{n=1}^\infty a_n$$ is convergent then $\lim\limits_{n\to\infty}a_n=0$. Proof. Let $s_n=\sum_{k=1}^n a_k.$ Then by the definition the limit $\lim_n ...
1
vote
2answers
57 views

Help with a proof I can't quite

Let $(F_j)^\infty_{j=1}$ be the sequence of Fibonacci numbers. For all $n \in \mathbb{N}$, $\sum\limits_{k=1}^{2n-1}F_kF_{k+1}=(F_{2n})^2$. I handled the base case quite well but couldn't go very ...
3
votes
1answer
74 views

Prove shuffled sequence $\{x_i, y_i\}$ converges $\iff \lim x_n = \lim y_n$ (Abbott p 49 q2.3.5)

Let $(x_{n})$ and $(y_{n})$ be given. Define $(z_{n})$ to be the shuffled sequence $(x_{1}.y_{1},\ x_{2},\ y_{2},\ x_{3},\ldots,x_{n}, y_{n},\ldots)$ . Prove that $(z_{n})$ is conv ergent $\iff ...
0
votes
1answer
21 views

Proving claims about sequences by induction?

I am learning how to prove claims about finite sequences right now. Can you help me prove or disprove the following claim? ...
0
votes
0answers
13 views

Prove that $p_{k +1}$ = $P_0$$(\frac{b}{d})$$^{k+1}$ doesn't converge by using partial sum of geometric series

Prove that $p_{k +1}$ = $P_0$$(\frac{b}{d})$$^{k+1}$ does not converge by using the partial sum of the geometric series if the conditions are not met. I know that the condition is d > b where they ...
3
votes
3answers
151 views

Fibonacci Sequence, Golden Ratio

I've been asked to show that $x_n \rightarrow L$ as $n \rightarrow \infty$ where $x_n = F_{n+1}/F_{n}$ for $n \in \mathbb{Z}^+$, where $F_n$ denotes the $n^{th}$ Fibonacci number. I am supposed to use ...
0
votes
1answer
25 views

Need help finding a formula for this sequence

A sequence $(x_j)^\infty_{j=0}$ satisfies $x_1=1$, and for all $m \ge n \ge 0 $ $x_{m+n}+x_{m-n} = \frac12 (x_{2m}+x_{2n})$. I have to find a formula for $x_j$ and then I can prove that later for ...
0
votes
1answer
62 views

Very complicated limit and trying to find convergence

I have no idea how to prove this: $\lim_{n \to \infty} \frac{2^2 \times4^2\times6^2\times\dots\times(2n)^2}{(1\times3)(3\times5)\dots((2n-1)(2n+1))} = \lim_{n \to \infty}\frac{2^2 ...
0
votes
2answers
56 views

Proof: $ M=\{x\in \Bbb{R}|\exists n \in \Bbb{N},\forall m \in \Bbb{N}( m \geq n\to a(m)\leq x)\}$ is bounded below

I need the proof of the following: Prop.: let be $a \in \Bbb{R}^\Bbb{N}$, and $a$ is bounded above, then: $$ M=\{x\in \Bbb{R}|\exists n \in \Bbb{N},\forall m \in \Bbb{N}( m \geq n \to a(m)\leq x)\} ...
7
votes
3answers
438 views

How to find the sum of $i(i+1)\cdots(i+k)$ for fixed $k$ between $i = 1$ and $n$?

I learned that $$\sum \limits_{i=1}^n i(i+1) = \frac{n(n+1)(n+2)}{3}$$ or in general $$\sum \limits_{i = 1}^n i(i+1)(i+2) \dots (i + k) = \frac{n(n+1)\dots (n+k+1)}{k+2}$$ From a mathematical ...
11
votes
2answers
1k views

Why is this allowed? (“Fourier's Trick”; finding the coefficients in a Fourier Series)

In my textbook (Introduction to Electrodynamics, D. Griffiths), we derive the equation for some strange potential function. Eventually, we get to this (for $n \in \mathbb{Z}^+$): $$ V_0(y) = ...
1
vote
2answers
74 views

Why does $\lim_{n \to\infty}a_{n+1} = \lim_{n\to\infty}a_n$?

Assume that $\{a_n\}$ is a convergent sequence. How to use the definition of a limit of a sequence to prove that
1
vote
2answers
265 views

Proof: Subsequence of n integers is divisible by n?

So I'm confused and stuck on how to approach this question. Any direction in the right path would be greatly appreciated. Let $n\in N$. Prove that any sequence of $n$ integers $a_1, a_2, ... a_n$ (no ...
1
vote
2answers
298 views

Use epsilon-N definition of a convergent sequence to prove that the series diverges

I'm stuck on this question - Use the ε-δ definition of a convergent sequence to prove that a series is divergent. I've attached the exact question but I have no idea what to do especially since it's ...
5
votes
1answer
295 views

Tricks. If $\{x_n\}$ converges, then Cesaro Mean converges (S.A. pp 50 2.3.11)

Show if $\{x_n\}$ is a convergent sequence, then the sequences given by the averages $\{\dfrac{x_1 + x_2 + ... + x_n}{n}\}$ converges to the same limit. (Not a duplicate) Let $\epsilon>0$ be ...
0
votes
1answer
30 views

$\exists L \in \Bbb{R}(\forall n \in \Bbb{N}(a^\frac{\lfloor b \cdot 10^n\rfloor}{10^n} \leq L))$?

Let be $a \in \Bbb{R}^{>0}, b \in \Bbb{R}$, $\exists L \in \Bbb{R}(\forall n \in \Bbb{N}(a^\frac{\lfloor b \cdot 10^n\rfloor}{10^n}\leq L ))$? I thought: if $a=1 \to 1^\frac{\lfloor b \cdot ...
0
votes
2answers
62 views

Proof of $\lim_{n\to\infty}\left(\frac{a_n+b_n}{2}\right)^n=\sqrt{ab}$

Let $a_n$ and $b_n$ two strictly positive sequences such that $$\lim_{n\to\infty}a_n^n=a>0\qquad \lim_{n\to\infty}b_n^n=b>0.$$ I need to prove that ...
1
vote
1answer
55 views

How to prove: If $a \to -\infty $ and $b$ is bounded from below by a constant $k\in\Bbb R^{>0}$, then the $a\cdot b\to -\infty$

I must proof the following, with $a: \Bbb{N} \to \Bbb{R}$ and $b: \Bbb{N} \to \Bbb{R}$ If $a \to -\infty\ (n\to\infty)$ and $b$ is bounded from below by a constant $k\in\mathbb R^{>0}$, then the ...
1
vote
1answer
60 views

Let $x$ be a real number. To prove…

Let $x$ be a real number. Define the sequence $(x_n)_{n\ge1}$ recursively by $x_1=1$ and $x_{n+1}=x^n+nx_n$ for $n\ge1$. Prove that, $$\prod_{n=1}^\infty \bigg(1-\dfrac{x^n}{x_{n+1}}\bigg)=e^{-x}$$ ...
0
votes
0answers
109 views

Using the root test twice.

Lets for example consider the following seires $\sum (something)^{n^2}$ Can I use the root test twice here and get $(something)$ then calculate the limite and then decide if the series is convergente ...