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

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3
votes
2answers
49 views

Indefinite Bessel integrals

I just ran into integrals of the Bessel type, but which are unfortunately indefinite integrals, such as $$ f(t)=\int \cos(\gamma\cos(\omega t))\cos(\omega t)\mathrm dt. $$ I'm conscious of the fact ...
1
vote
2answers
92 views

How to show that such a sequence exist?

Suppose that a non-constant function $\gamma : \mathbb{R} \to \mathbb{R}$ is $T$ -periodic for some $T \neq 0$. This exercise shows that there is a smallest positive number $T_0$ such that $\gamma$ ...
3
votes
1answer
34 views

Radius Of Convergence Confusion

Given the series $\sum^{\infty}_{n=1} \frac{n!z^n}{n^n}$ find the radius of convergence. Well, I know that if the following Lemma holds: Lemma Given the series $\sum^{\infty}_{n=1} c_nz^n$ where ...
1
vote
3answers
117 views

Example of a sequence that is not Cauchy

Question: Find an example such that $|x_{k+1}-x_k|\to 0$, but the sequence $x_k$is not Cauchy. Any help would be appreciated.
0
votes
1answer
42 views

Convergence or Divergence of infinite series

$\sum_{n=1}^{\infty} {z^n\over(n + 1)!}$ So I got $\sum_{n=0}^{\infty} {z^n\over(n + 1)!} - 1$. I understand that if $|z|<1$ then $z^n$ approaches 0 as n approaches infinity. This leaves me ...
0
votes
0answers
46 views

How is the sequence $A_n = \frac{1 \times 3 \times 5 \times \cdots \times (2n-1)}{2 \times 4 \times 6 \times \cdots \times (2n) }$ calculated?

Note the sequence $$A_n = \frac{1 \times 3 \times 5 \times \cdots \times (2n-1)}{2 \times 4 \times 6 \times \cdots \times (2n)}$$ Simple, question, sorry if it's too simple..How is this read? What ...
2
votes
3answers
173 views

Show that $n^{(1/n)}$ is eventually decreasing

Show that $$ a_n=n^{1\over n}$$ is eventually decreasing. I am not allowed to used derivatives and I have been trying for a while. I graphed it and it is decreasing if $n>2$, but how can you show ...
2
votes
1answer
29 views

finding a suitable region for a series of functions to be analytic

Let $f_n(z) = \frac{(2z-1)^n}{n} $. I want to find some suitable $A \subset \mathbb{C}$ such that $\sum_{n=1}^{\infty} f_n(z) $ is analytic on $A$. Well, first of all $\sum f_n (z) $ must be ...
0
votes
1answer
24 views

“Nice” lower bound for the exponential partial sum

Let $f_n = \sum _{k=0} ^{n-1} \frac{(-1)^k(\ln a)^k}{k!}, 0<a<1$. My intuition says this converges very rapidly to $1/a$, so I need to compare it to a nice sequence. My candidate was $b_n = ...
1
vote
0answers
77 views

Find the limit of the sum of normal pdf's

How to find the limit of the sum of Gaussian probability distribution functions? $$ \lim_{N \to \infty} \sum_{k=1}^N \frac{1}{\sigma \sqrt{2 \pi k}} \exp\left(-\frac{(N-k\mu)^2 ...
2
votes
1answer
31 views

series converges but not absolutely

Let $a_n = \frac{ i^n}{n} $ where $i = \sqrt{-1} $. It is clear that $\sum |a_n| = \sum \frac{1}{n} $ diverges. But, how about $\sum a_n $, does it converge ? Obviously, using the ratio test, gives ...
25
votes
4answers
347 views

How many ways are there to pile $n$ “$1\times 2$ rectangles” under some conditions?

A friend of mine taught me the following question. He said he created the question by himself and conjectured the answer, but couldn't prove it. Though I've tried to solve the question, I've been ...
0
votes
1answer
27 views

How to calculate the recursive form of an explicit sequence?

I'm in trouble with this questions. $a_n=5^n$ where $n=1,2,3,...$ $a_1=5^1=5,\, a_2=5^2=25,\, a_3=5^3=125$ I always try to resolve such questions by using $a_n-a_{n-1}$ But this time I can't ...
1
vote
1answer
51 views

Proving the limit of $\frac{n!}{10^{n}}$ using definitions

$\cdot \lim \limits_{n \to \infty} \frac{n!}{10^n} = \frac{10!}{10^{10}} * (\frac{n!}{10^n})$ for all $n \ge 11$ So we must find $N(M)$ such that $\lvert \frac{10!}{10^{10}} * (\frac{n!}{10^n}) ...
-4
votes
1answer
54 views

Find {${a_n}$} such that $\sum_{n}a_n $ is convergent, but $\sum_{n}a_n^2 = \infty $. [closed]

(a) Find {${a_n}$} such that $\sum_{n}a_n $ is convergent, but $\sum_{n}a_n^2 = \infty $. (b) You can find a sequence or {${a_n}$} of positive numbers satisfying (a) ?
0
votes
3answers
71 views

What is the limit when n goes to infinity? [closed]

How to find the limit of $\frac{(-3)^n}{n!}$ when $n$ goes to $\infty$ by the squeeze theorem or any other method?
2
votes
1answer
51 views

$\{a_n\}$ has no convergent subsequence if $|a_n - a_m| \ge \epsilon$.

\Let $\{a_n\}$ be a sequence such that for some $\epsilon >0$, $$ \lvert a_n-a_m\rvert \ge \epsilon$$ for all $n\neq m$. Prove that $a_n$ has no convergent subsequence. My thoughts on this is to ...
23
votes
1answer
446 views

How should I calculate $\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+…+n^n}{n^n}$

How should I calculate the below limit $$\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+...+n^n}{n^n}$$ I have no idea where to start from.
0
votes
1answer
72 views

Prove the following statement about Cauchy sequences… [duplicate]

...without using the fact that a Cauchy sequence of real numbers converges: If a subsequence of a Cauchy sequence converges, then the Cauchy sequence converges.
0
votes
4answers
44 views

proof of limit involving factorials and exponents

$ \cdot \lim \limits_{n \to \infty}\frac{10^n}{n!} $ I know intuitively that this is zero but I'm not sure how to prove this. Can I use an inequality? Maybe $\frac{10^n}{n!} \le \frac{1}{n!}$ when ...
15
votes
3answers
493 views

How show that $|a_{n}-1|\le c\lambda ^n,\lambda\in (0,1)$

I'm looking for proof that Let $a_{0},a_{1}>0$,and such $$a_{n+2}=\dfrac{2}{a_{n+1}+a_{n}}$$ show that: there exist $\lambda,c>0$ such $$|a_{n}-1|\le c\lambda ^n,\lambda\in (0,1)$$ I tried ...
1
vote
2answers
33 views

proving the divergence of a sequence using definitions

Use appropriate definitions to show that, $\cdot \lim \limits_{n \to \infty} 10 -\sqrt{n} = -\infty $ So we need to find a function N(M) for each $M>0$ such that $\lvert 10 - \sqrt{n}\rvert > M ...
3
votes
2answers
83 views

Is it true that $\sum_\limits{n = 0}^\infty \dfrac{a_n}{n}$ is also convergent [duplicate]

Let $\{a_n\}_{n = 0}^\infty$ be a sequence of real numbers such that the series $\sum_\limits{n = 0}^\infty |a_n|^2$ is convergent. Is it true that $\sum_\limits{n = 0}^\infty \dfrac{a_n}{n}$ is also ...
-1
votes
1answer
43 views

Is $\sum_{n=1}^{\infty }\tan^{-1}\Big(\frac{1}{n^{100}}\Big)=\frac{\pi}{4}$ [closed]

I ask if the following equality is true or not $$\sum_{n=1}^{\infty }\tan^{-1}\Big(\frac{1}{n^{100}}\Big)=\frac{\pi}{4}$$
0
votes
1answer
55 views

Sum of $i$ times $(i-1)^\text{th}$ Fibonacci Number [closed]

Consider the expression $$\sum\limits_{i=1}^n i \cdot F_{i-1}$$, where $F_{0}=0, F_{1}=1, F_{2}=1, F{3}=2,$ etc. Is there a closed formula for this? If so, find it.
1
vote
1answer
25 views

Prove that $\sum_{k = 0}^d 2^k \log(\frac{n}{2^k})= 2^{d+1} \log (\frac{n}{2^{d-1}}) - 2 - \log n$

Am trying to prove that $$\sum_{k = 0}^d 2^k \log(\frac{n}{2^k})= 2^{d+1} \log (\frac{n}{2^{d-1}}) - 2 - \log n$$ I don't know how to go about it. Any ideas? I have tried to change the $2^k \log ...
1
vote
2answers
59 views

Sequences Tending to Infinity Proof

Suppose that $(a_n)$ tends to infinity, and $(b_n)$ is greater than or equal to $-100$ for all $n$ in the naturals. Is it necessarily true that ($a_n + b_n$) tends to infinity? Prove your answer is ...
2
votes
1answer
70 views

Sequence diverging to infinity is not bounded above?

Problem: Prove that $(a_n)$ -> $\infty$ implies that $(a_n)$ is not bounded above. My attempt: Let $C>0$ be arbitrary. Let $(a_n)\to\infty$ as $n\to\infty$. By definition, $\forall ...
0
votes
3answers
141 views

Can a unbounded sequence have a convergent sub sequence?

I have been using the Bolzano-Weierstrass Theorem to show that a sequence has a convergent sub sequence by showing that it is bounded but does that mean that if a sequence is not bounded then it does ...
1
vote
2answers
58 views

$\sum_{j=3}^\infty \frac{1}{j(\log(j))^3}$ converges or diverges?

Showing that $\sum_{j=3}^\infty \frac{1}{j(\log j)^3}$ diverges or converges How would it converges or diverges. I thought about using comparison test buy $\frac{1}{j}$ is bigger than the sum. ...
0
votes
3answers
45 views

Showing $a_{j+1}=\frac{1}{2}(\frac{4}{a_j}+a_j)$ is convergent

How to show the following sequence is decreasing and bounded below. Let $a_1>2$ for $j \ge 1$ define $a_{j+1}=\frac{1}{2}(\frac{4}{a_j}+a_j)$ I am trying to show it is decreasing First I did ...
3
votes
2answers
88 views

$\int_0^\infty \frac{x}{e^x-1} dx$ with series

I've been looking at the following integral, and trying to solve it through series: $$\int_0^\infty \frac{x}{e^x-1} dx$$ I tried expanding out $e^x-1$: $$\int_0^\infty ...
2
votes
2answers
44 views

Radius Of Convergence of the series

I want to calculate the radius of convergence of the series $\sum^{\infty}_{n=0} x^{n!}$. I've tried the ratio test to no avail, obtaining: $\frac{|x^{(n+1)!}|}{|x^{n!}|}= |x^{(n+1)!-n!}| = ...
0
votes
3answers
68 views

If $\sum_{n=1}^\infty a_n$ converges, prove that $\lim_{n\to \infty} (1/n) \sum_{k=1}^n ka_k = 0$.

the series $a_n$ is any arbitrary converging series. My thought process was that the $1/n$ will definitely go to zero as n approaches infinity; however, the series $k*a_k$ seems to approach infinity ...
1
vote
2answers
49 views

Show that $\sum_{j=1}^{\infty}\frac{\sqrt{j+1}-\sqrt{j}}{j+1}$ is convergent or divergent.

How would I show that the following series converges or diverges. $\sum_{j=1}^{\infty}\frac{\sqrt{j+1}-\sqrt{j}}{j+1}$ I am not sure what is the best test to show that it converges. Apparently it ...
7
votes
3answers
124 views

The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)\cdots(1+x^{100})?$

What is the coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)\cdots (1+x^{100})?$ I manually expanded $(1+x)(1+x^2)(1+x^3)...(1+x^{10})$ and calculated the coefficient of $x^9$ as $8$ ...
0
votes
1answer
21 views

This is the first half of proving Bolzano-Weierstrass theorem

Just making sure I'm on the right track so far Every bounded infinite set of real numbers has at least one accumulation point Pf: Let S be a bounded set. Since S is bounded, there are real numbers ...
4
votes
1answer
64 views

On evaluating the Riemann zeta function, including that $\zeta(2)\gt \varphi$ where $\varphi$ is the golden ratio

A week ago, I got the following : For a positive integer $k$, using Cauchy–Schwarz inequality, $$\left(\sum_{n=1}^{\infty}\frac{1}{n^k(n+1)^k}\right)^2\lt ...
2
votes
1answer
35 views

Prove $(\sum\limits_k a_k b_k)^2 \leq \sum\limits_k b_k a_k^2 \sum\limits_k b_k$ using Cauchy Schwarz

Let $a,b$ be two $n$ dimensional vectors, we want to show that $(\sum\limits_k a_k b_k)^2 \leq \sum\limits_k b_k a_k^2 \sum\limits_k b_k$ Recall the Cauchy Schwarz inequality is given as $|\langle ...
5
votes
0answers
89 views

Double sum involving $\cos$

I ran across a double sum and was wondering if someone may be adept at evaluating it. I must admit that my double summation skills could be better, and I am always ready to learn more. Show that: ...
0
votes
2answers
34 views

Convergent sequence: If $\lim a_n\ne0$ then $|a_n|>\delta$

How do I prove that if $(a_n)$ is a Cauchy sequence and $\lim\limits_{n \to \infty}a_n \neq 0 $ then there exists $n_0 \in \mathbb{N}$ and $\delta >0$ such that $|a_n| > \delta $ for all $n > ...
0
votes
1answer
38 views

How to prove $\log n$

$$n, \frac n2,\frac n4,\frac n8,\frac n{16}\dots 1$$ In this series $n$ to $1$, there are $x$ number of steps. Then how to prove (Can we prove?) $\log_2 n = x$
0
votes
2answers
31 views

Prove implication all Cauchy sequences have a limit $\to$ all monotone increasing bounded above sequences converge$

So, we assume that all Cauchy sequences converge. How can we deduce ''all monotone bounded above increasing sequences converge'' from that? Any hints would be appreciated.
0
votes
0answers
26 views

Find the next 5 terms of a G.P 5 and 25

Find the next $5$ terms of a G.P $5$ and $25$ So I solved it like this, $5^1= 5$ $5^2= 5\times5= 25$ $5^3= 25\times5=125$ $5^4= 125\times5=625$ $5^5= 625\times5=3125$ Is the method correct? Or ...
2
votes
2answers
57 views

$\sum_{k=n}^{\infty}\left(n-k\right)e^{-\lambda}\frac{\lambda^{k}}{k!}= ?$

Could you please help me. How do I sum the following: $$\sum_{k=n}^{\infty}\left(n-k\right)e^{-\lambda}\frac{\lambda^{k}}{k!}$$ If the summation had started at 0, then it would be simply an ...
3
votes
1answer
68 views

How to find $\lim_{x\to +\infty}{\left(\sqrt{\pi}x-\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+2}}{2n!(2n+1)(2n+2)}\right)}$?

$$\lim_{x\to+\infty}{\left(\sqrt{\pi}x-\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+2}}{2n!(2n+1)(2n+2)}\right)}$$ This limit comes from the following integral, which I expand it into Taylor series. ...
-1
votes
1answer
38 views

is there a name for this series $p_{n,j}=\sum_{k=1}^n (k^2\pi)^j$?

Is there a name for this series? $$p_{n,j}=\sum_{k=1}^n (k^2\pi)^j$$ Mathematica 10.0 does not produce closed-form expression.
1
vote
1answer
24 views

Difference of Riemann sums

Suppose $f(x)$ is a non-negative function such that $\int_{\mathbb{R}}f(x) \ dx=1$. Define $p_{n,k}$ as follows: $$ p_{n,k}= \int_{\frac{k}{n}}^{\frac{k+1}{n}} f(x) \ dx $$ We know that ...
1
vote
1answer
31 views

Using the Monotone Convergence Theorem to prove convergence of a recursively defined sequence.

Say $x_n=2+\sqrt{x_{n-1}-2}$ and $x_0\geq2$ for $n\in\mathbb{N}$. Use the Monotone Convergence Theorem to prove that $x_n\to2$ or $x_n\to3$ as $n\to\infty$. Specifically, how do I prove that $x_n$ is ...
4
votes
2answers
117 views

My incorrect approach solving this limit. What am I missing?

I'm doing this limit: $$-1/2\lim_{n\to\infty}\,{\frac {n\left( 2\,{{\rm e}^{2}} \left( {\frac {n+1}{n}} \right) ^{-2\,n}n-1-2\,n \right)}{n+1} } $$ I arrived to this from a bigger expression, and ...