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

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0
votes
1answer
19 views

Proof of sum in an inequality

I was having hard time solving this one, any help will be greatly appreciated. prove that: $$ {39\over e^2}\le\sum_{n=1}^\infty {4n^2-1\over e^n}-{3\over e}\le{54\over e^2} $$
0
votes
2answers
24 views

Error in a Maclaurin series

I'm having trouble figuring out what I have to do with this question. "Using Taylor's theorem, determine the largest positive real value r for which we can guarantee that the Maclaurin polynomial ...
4
votes
2answers
49 views

Is it true that: $|a_{n+1} - L| < |a_{n} - L| \forall n \in \mathbb{N} \implies \lim \limits_{n \to \infty} a_{n} = L ?$

If $a_{n}$ is a sequence and $|a_{n+1} - L| < |a_{n} - L|, \forall n \in \mathbb{N} $, then clearly the sequence $s_{n} = |a_{n} - L|$ converges (it's decreasing and bounded by $0$). Does it ...
0
votes
0answers
19 views

Closed form for series involving harmonic numbers

Is there a closed form for this series values: $$ \sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(n)}}{k} $$ where $$ H_k^{(n)}=\sum_{i=1}^k \frac{1}{i^n} $$ Thanks!
2
votes
3answers
104 views

Determining the limit of this series

$$ \sum_{n=0}^\infty \frac{(-2)^n + 2^{3n}}{3^n4^n} = $$ $$ \sum_{n=0}^\infty \frac{(-1)^n2^n}{3^n4^n} + \sum_{n=0}^\infty \left(\frac{2}{3}\right)^n = $$ $$ \sum_{n=0}^\infty (-1)^n\frac{1}{6^n} + ...
1
vote
0answers
31 views

Is the Ulam Spiral just a coincidence?

I was messing around with the Ulam spiral because I was a little skeptical on it having any actual relevance. I noticed that if you lay out the spiral and then circle all the even numbers, it displays ...
1
vote
0answers
23 views

Fibonacci-related infinite sum

Prove that $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}}=\frac{\sqrt{5}}{4}\theta_2^2\bigg(\frac{3-\sqrt{5}}{2}\bigg)$$ and $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}+F_{2k-1}}=\frac{(2k-1)\sqrt{5}}{2F_{2k-1}}$$ ...
-2
votes
1answer
25 views

Subsequence $x_{nk}$ tend to ∞? [on hold]

We have a sequence $x_n$ where $n\geq1$ so that $x_n$->∞.
-2
votes
1answer
42 views

Existence of divergent series $\sum_{n=1}^ \infty a_n$ of real numbers whose partial sums are bounded and $\lim (na_n)=0$ [on hold]

Does there exist a sequence $(a_n)$ of real numbers such that $\lim_{n \to \infty} (na_n)=0$ , the partial sums of $\sum_{n=1}^ \infty a_n$ are bounded , but $\sum_{n=1}^ \infty a_n$ is divergent ? ...
0
votes
1answer
21 views

Could the Hamel basis of $\mathbb{R^Z}$ be the set $\mathbb{R^Z}-{\mathbf{\{0\}}}$?

This is the follow up question to this question (*) According to page 2 of this link 1 and this link 2, $\mathbb{R^Z}$ (which is referred as $\mathbb{R^\infty}$ in link 1) has elements of the ...
2
votes
1answer
44 views

Let $a_n>0$ for $n \geq 1$ and let series: $\sum_{n=1}^{\infty}a_n$ diverge. Let $S_n=a_1+a_2+…+a_n > 1$ for $n \geq 1$

Prove that the series: $$\sum_{n=1}^{\infty}\frac{a_{n+1}}{S_n \ln S_n}$$ diverges and the series : $$\sum_{n=1}^{\infty}\frac{a_{n}}{S_n \ln^2 S_n}$$ converges. (Using the famous criteria I ...
1
vote
1answer
77 views

Is $\mathbb{R^Z}$ or its elements countable?

Continue on the self study on infinite vector spaces. According to this link, $\mathbb{R^Z}$ has elements of the following form: $$(y_k)_{k\mathbb{\in Z}}=(\dots y_{-1},y_0,y_{1}\dots)$$ Or more ...
-6
votes
0answers
19 views

Ples help with this series [on hold]

1/(n^(3) +2n)^(1/2) it converge or diverge
2
votes
1answer
81 views

Calculate $\lim_{n \to \infty}(\sin nx)^\frac{1}{n}$

I know that $$(n)^\frac{1}{n} \to 1$$ and $(a)^\frac{1}{n} \to 1$ (with $a \in \mathbb{R+}$). However, I was wondering what can be said about $$\lim_{n \to \infty }(\sin nx)^\frac{1}{n}$$ and, more ...
2
votes
6answers
100 views

Calculate the sum of three series which may be telescoping

Let $$\sum_{n=1}^\infty \frac{n-2}{n!}$$ $$\sum_{n=1}^\infty \frac{n+1}{n!}$$ $$\sum_{n=1}^\infty \frac{\sqrt{n+1} -\sqrt n}{\sqrt{n+n^2}}$$ I have to calculate their sums. So I guess they are ...
-3
votes
1answer
35 views

Does $\sum a_n$ converge if $a_1 = 1$ and $a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n$ [on hold]

Let $$a_1 = 1$$ and $$a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n.$$ Consider $$\sum a_n.$$ How do I calculate if the series converges? The definition by recurrence troubles me a lot.
2
votes
0answers
39 views

$ \sum_{n=1}^{\infty}\frac{1}{(an^2+b)^k} $ equals $q_0+q_1\pi+\dots+q_k\pi^k$ for nonzero rational coefficients

is it possible to find $a,b\in\mathbb{Z}$ such that for every $k\in\mathbb{N}$ the sum $$ \sum_{n=1}^{\infty}\frac{1}{(an^2+b)^k} $$ equals $q_0+q_1\pi+\dots+q_k\pi^k$ for some nonzero ...
3
votes
3answers
112 views

Determine if a series defined by cases is convergent and calculate the sum

Consider $\sum_{n=1}^\infty a_n$, where $a_n$ is $$3^{-n}$$ if $n$ is even and $$\ln \frac{(n+2)(n+1)}{n(n+3)}$$ if $n$ is odd. I have to say if it is convergent and calculate its sum, but the ...
6
votes
1answer
62 views

the series: compute $ \sum_{n=1}^{\infty}\frac{1}{(4n^2-1)^4} $

Compute $$ \sum_{n=1}^{\infty}\frac{1}{(4n^2-1)^4} $$ the result is $\frac{\pi^4+30\pi^2-384}{768}$, so I'm sure the sums $\sum\frac{1}{n^2}$ and $\sum\frac{1}{n^4}$ should appear in the solution. ...
1
vote
2answers
58 views

Find $f$ so that $\int_{1}^{\infty}f(x)dx$ exists, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist? [duplicate]

I need help finding an example of a function such that $\int_{1}^{\infty}f(x)dx$ converges, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist. I was trying to find examples of functions ...
0
votes
1answer
32 views

Sigma sign problem from Spivak's calculus text ch 2-2

I need to find a formula for $$ \sum_{i=1}^n (2i-1)^2 = 1^2 + 3^2 + \cdots + (2n-1)^2 $$ This problem is contained in Spivak's calculus ch2-2. I know that: $$ \sum_{i=1}^n i^2 = ...
4
votes
5answers
89 views

Calculate $\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$

How can I calculate the following limit? I was thinking of applying Cesaro's theorem, but I'm getting nowhere. What should I do? $$\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$$
3
votes
3answers
75 views

How to demonstrate that a sequence has subsequences converging in all points of $[0,1]$?

The sequence is: $S = (\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{2}{4},\frac{3}{4}...)$ I want to prove that $\forall \alpha \in [0,1]$, there exists a subsequence of $S$ which converges ...
2
votes
1answer
41 views

For which values of $x$ is the following series convergent: $\sum_0^\infty \frac{1}{n^x}\arctan\Bigl(\bigl(\frac{x-4}{x-1}\bigr)^n\Bigr)$

For which values of $x$ is the following series convergent? $$\sum_{n=1}^{\infty} \frac{1}{n^x}\arctan\Biggl(\biggl(\frac{x-4}{x-1}\biggr)^n\Biggr)$$
1
vote
1answer
37 views

Uniform convergence of a recursive increasing function

Let $\phi:[0,+\infty)\rightarrow\mathbb{R}$ an increasing, continous function such that $\frac{1}{2}\leq\phi(x)\leq1$ for all $x$. Let $f_0:[0,+\infty)\rightarrow\mathbb{R}$ an increasing function ...
8
votes
7answers
665 views

What mistake have I made when trying to evaluate the limit $\lim \limits _ {n \to \infty}n - \sqrt{n+a} \sqrt{n+b}$?

Suppose $a$ and $b$ are positive constants. $$\lim \limits _ {n \to \infty}n - \sqrt{n+a} \sqrt{n+b} = ?$$ What I did first: I rearranged $\sqrt{n+a} \sqrt{n+b} = n \sqrt{1+ \frac{a}{n}} \sqrt{1+ ...
1
vote
0answers
24 views

Estimate from above $\sum_{m=1}^{n-1}\frac{1}{n^\alpha-m^\alpha}$

Find an upper bound for $$\sum_{m=1}^{n-1}\frac{1}{n^\alpha-m^\alpha}$$ with $\alpha>1$. I do not know where to start but, for example, if $\alpha=1$ the previous sum is linked to Harmonic numbers ...
0
votes
1answer
32 views

With the aid of series, show that if $f(z)=\frac{\operatorname{cos}z}{z^2-(\pi/2)^2}$, then $f$ is an entire function.

Prove that if $$f(z)= \begin{cases} \frac{\operatorname{cos}z}{z^2-(\pi/2)^2}, & \text{when} \; z\neq \pm \pi/2, \\ -\frac{1}{\pi}, & \text{when} \; z=\pm \pi/2, \end{cases} $$ then $f$ is ...
-3
votes
1answer
22 views

Show, directly from the definition, that the following series is convergent. [on hold]

Using the definition of a convergent series, how do you show that the series $\sum_{n=1}^{\infty} (\frac{-2}3)^n $ converges.
0
votes
2answers
24 views

Recurrence sequences with two initial condition: how do I calculate the limit?

I've done some exercises with recurrence sequences with one initial condition. So, now that I'm attempting one exercise with two initial conditions I'm confused. Could you show me what to do? Let ...
3
votes
1answer
34 views

Is it true that $\sum_{n=1}^\infty \dfrac{Var(Y_n)}{n}<\infty$?

Suppose that $\{X_n\}$ is an i.i.d. sequence of random variables with $E|X_1|<\infty$.Define $Y_n=X_nI_{\{|X_n|<n\}}$ for all $n\geq1$. Is it true that $\sum_{n=1}^\infty ...
0
votes
1answer
42 views

Limit of $a_{n+1}= \frac{n}{n+1} a_n$

I think that this sequence $$a_{n+1}= \frac{n}{n+1} a_n$$ can be rewritten as $$a_n= \frac{1}{n+1}a_0.$$ Therefore the limit should be $0$. But my proof by induction turns out wrong. Is my idea ...
2
votes
4answers
90 views

How to show convergence of $\sum_{n=1}^{\infty}\log(1 + \frac{1}n)$?

I am trying to prove whether \begin{equation*} \sum\limits_{n=1}^{\infty}\log(1 + \frac{1}n) \end{equation*} converges or diverges, but none of the normal tests (nth test, p test, etc. ) seem to ...
1
vote
4answers
66 views

Find the limit of the sequences: $a_{n+1}=3a_n - n + 1$ and $(a_n)^\frac{1}{n}$ with $a_0 > 0 $

Let $a_0 > 0 $ and $$a_{n+1}=3a_n - n + 1.$$ I have to find its limit. I have also to find the limit of $(a_n)^\frac{1}{n}$. But this seems even more complicated. For the first part I've used the ...
2
votes
3answers
59 views

Prove that a certain sequence is increasing and find its limit: $a_1 = 1$ and $a_{n+1}=n(1+\ln a_n)$ (and $(a_n)^\frac{1}{n}$)

Let $a_1 = 1$ and $$a_{n+1}=n(1+\ln a_n).$$ I have to find its limit. I want to prove that it is increasing for starters, but I'm already stuck. What should I do? I have also to find the limit of ...
1
vote
1answer
36 views

Find the value of $\sum^{n-1}_{m=1}\left(\frac{1}{n-m}+\frac{1}{n+m}\right)$.

Find the value of $$\sum^{n-1}_{m=1}\left(\frac{1}{n-m}+\frac{1}{n+m}\right)$$ I used WolframAlpha obtaining $$\psi^{(0)}(2n)-\frac{1}{n}+\gamma$$ where $\gamma$ is the Euler-Mascheroni constant and ...
0
votes
2answers
80 views

How to prove that the sum of a convergent geometric series of the form $1 + r + r^2 … + r^n > 1/2$?

I am trying to prove that the sum of a convergent geometric series of the form \begin{equation*} 1 + r + r^2 .... + r^n > \frac{1}{2} \end{equation*} but I have no idea how to go about this. ...
1
vote
2answers
43 views

Establishing a trigonometric identity for $n\in\mathbb{N}$

The original problem was showing that this infinite sum converges to $\tan\theta$: $$\sum_{n=1}^\infty \frac{\tan\dfrac{\theta}{2^n}}{\cos\dfrac{\theta}{2^{n-1}}}$$ One hint was given: the series ...
2
votes
1answer
24 views

Radius of convergence of a power series with $a_n$ convergent

Let $\{a_n:n \geq 1\}$ be a sequence of real numbers such that $\sum_{n=1}^{\infty} a_n$ is convergent and $\sum_{n=1}^{\infty} |a_n|$ is divergent. Let $R$ be the radius of convergence of the power ...
3
votes
2answers
25 views

Determine whether series converges or diverges

$$\sum_{n=1}^{\infty}\frac{\sin\left(\frac{5\pi}{3}n\right)}{n^{\frac{5\pi}{3}}}$$ Hello, I thought about using Squeeze Theorem but the 5π/3 threw me off. Thanks in advance.
0
votes
2answers
34 views

Can one prove divergence by showing a series has at most one solution for an=0?

Say I have any series, I would think it was enough to show that this series equals 0 at most once to prove it diverges. My logic is, For a series: $\sum a_n →∞$, and diverges, if $a_n≠0$ for $n→∞$ ...
2
votes
3answers
70 views

Math about Geometric series

In a geometric series, the sum of $1^{st}$ term $+$ $2^{nd}$ term $+$ $3^{rd}$ term $= 38$, the sum of $2^{nd}$ term $+ 4^{th}$ term $= 17 \frac{1}{3}$; how to calculate the common ratio? ( it is ...
1
vote
0answers
44 views

Uniformly convergence question

I have the following exercise: Let $\varphi:[0,+\infty)\to \mathbb{R}$ an increasing continuous function that satisfies that $1/2\leq \varphi(x) <1$ for all $x>0$. Let $f_0:[0,+\infty)\to ...
3
votes
0answers
50 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
1
vote
0answers
16 views

Analytic function with alternating taylor series

Is there a characterization of the sequences of non-negative real numbers $\{a_n\}_{n=0}^{\infty}$ such that $$ f(x)=\sum_{n\geq 0}a_n (-x)^n $$ converges absolutely for all $x\geq 0$? It's not hard ...
2
votes
4answers
96 views

Compute $\lim_{n \to \infty} \left(\frac{1}{\sqrt{n^3+1}} + \frac{1}{\sqrt{n^3+4}} + \cdots + \frac{1}{\sqrt{n^3+n^2}}\right)$

How do I evaluate the following limit? I guess I should do a comparison, but I've got no clue about what to do. Could you give me a hand? $$\lim_{n \to \infty}\left( \frac{1}{\sqrt{n^3+1}} + ...
0
votes
2answers
42 views

Calculate: $\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$ and $\lim_{n \to \infty} \frac{10^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$

I have to evaluate the following limits (which are similar). However, I don't know how to evaluate them. Could you give me a hand? $$\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln ...
5
votes
3answers
112 views

How can I calculate $\lim_{n \to \infty} (1 + \frac{1}{n!})^n$ and $\lim_{n \to \infty} (1 + \frac{1}{n!})^{n^n}$?

How do you calculate the following limits? $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^n$$ $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^{n^n}.$$ I really don't have any clue ...
-4
votes
2answers
62 views

Why is $n^\sqrt n - 2^n \to - \infty$ and $\sqrt n ^n - 2^n \to +\infty$ [on hold]

Can you explain technically why the following limits are correct? $$n^\sqrt n - 2^n \to - \infty$$ and $$\sqrt n ^n - 2^n \to +\infty$$
-1
votes
2answers
51 views

How do I calculate the following limit? $\lim_{n \to \infty} n ((8 + \sin (2^\frac{1}{n}))^\frac{1}{3} -2)$

How do I calculate the following limit? I'm short on ideas for this one: $$\lim_{n \to \infty} n ((8 + \sin 2^\frac{1}{n})^\frac{1}{3} -2)$$