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

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2answers
25 views

Convergence of ${a_n}$ = $\sum _1^n \frac{1}{n^{1+\alpha}}$ [duplicate]

Does the series ${a_n}$ = $\sum _1^n \frac{1}{n^{1+\alpha}}$ converge for all $\alpha$ > 0?
0
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0answers
29 views

Radius of convergence of the series-power series

Can anyone help me to check whether my solution is correct because we are not provided with the solutions,but I want to ensure what I did is correct. Thanks for your help! (a)$\sum_{n=1}^\infty 5^n ...
1
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2answers
103 views

Two convergent subsequences and their limits

$\{a_n\}$ is a sequence. I'm asked to verify the following statement: "If $\{a_{2n}\}$ and $\{a_{3n}\}$ converge then $$\lim_{n\to\infty}a_{2n}=\lim_{n\to\infty}a_{3n}$$ I think this is not true, but ...
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0answers
22 views

How can I show that the series below converges/or diverges? [on hold]

I don't know how to approach this problem. I would appreciate any ideas/help. [(1/2)*(1/2)]/(9*7*25*1!) +[(1/2)(3/2)(3/2)]/(11*9*49*2!) +[(1/2)(3/2)(5/2)*(5/2)]/(13*11*81*3!) + ...
1
vote
1answer
20 views

Convergent sums and rate of decay

True or False: If $a_n\in\ell^1,$ then $\overline{\lim}n a_n<\infty$ (i.e. $a_n=O(\frac{1}{n})$) Edit: My intuition says the answer should be positive.
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0answers
30 views

Calculating age with decreasing year values

This is my first question on math.stackexchange, with everything this entails. This question is about the perceived duration of every year as one ages. We will call $Y_0$ the perceived duration of ...
4
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3answers
36 views

Proving statement about convergent sequence $(a_n)$ and the sequence $(\max\{a_n,a_n^2\})$

Suppose $(a_n)$ is a sequence and $\lim_{n\to\infty} a_n = a$ and let $(b_n)=(\max\{a_n,a_n^2\})$. I have to prove/disprove that: If $a>1$ then $\lim_{n\to\infty} b_n = a^2$ If $a=1$ then ...
0
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1answer
30 views

Accumulation point is a limit of some sequences in first-countable space with Axiom of Choice.

I have a statement which says that Let $X$ be a first-countable space and $E$ be a subset of $X$. If $p$ is an accumulation point of $E$, then there exists a sequence in $E$ that converges to ...
4
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1answer
61 views

Help in finding the sum of the series

$$\sum_{n=1}^\infty \frac{1}{n^4+n^2+1}$$ I tried breaking into factors but it is not telescoping. $$\frac {1}{(n^2+n+1)(n^2-n+1)} = \frac {1}{2n} \left(\frac {1}{n^2-n+1} - \frac ...
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1answer
25 views

double root and newton method, a problem on solved exercise?

$f(x)$ in $x= \alpha$ has double roots and define in $\alpha$ neighbor. if the sequence $\{x_n\}$ for solving $f(x)=0$ calculated by newton methods the following is correct. ($a$ and $b$ is plasced ...
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0answers
13 views

Show the equivalence of two infinite series over Bessel functions

The following sums pop up in diffraction theory and are related to Lommel's function of two variables. Let $u,v\in\mathbb{R}$. I claim that $$\sum_{n=0}^\infty i^n \left ( \frac{u}{v} \right )^n ...
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1answer
24 views

Divergent series of independent RV

I'm trying to prove that if $\{X_n\}_{n=1}^{\infty}$ is a sequence of independent random variables with the same distribution and $P(X_1 \neq 0)>0$, then the series $\sum_{n=1}^{\infty} X_n$ is ...
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1answer
31 views

Alternating series error estimatation theorem for $\sum_1^{\infty} (-1)^{n+3}\frac{n}{n^2+8}$

$$\sum_1^{\infty} (-1)^{n+3}\frac{n}{n^2+8}$$ Here is my question: Using the alternating series error estimation theorem, what is the smallest number of terms needed to estimate the entire sum with ...
-2
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2answers
34 views

Explain why the following sums of a harmonic series is greater than or equal to 1/2. [on hold]

The (non-geometric) series $\frac{1}{2} + \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \cdots$ is called the harmonic series. a) Explain why each of the following sums is greater than or equal to 1/2. ...
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1answer
34 views

Prove or disprove: if the limit of $a_{n}$ goes to $\infty$, then $a_{n}$ has a minimum number.

Prove or disprove: if the limit of $a_{n}$ goes to $\infty$, then $a_{n}$ has a minimum number. Well I know from the definition of a series that goes to $\infty$ that there is a number $K>0$ that ...
2
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2answers
66 views

Prove or Disprove: If $a_n$ was positive and its limit is positive then the limit of the $\sqrt[n]{a_n}$ is $1$

If $a_n$ was positive and its limit is positive then the limit of the $\sqrt[n]{a_n}$ is $1$. I think it has to do with multiplication to start here,but I couldnt figure it out.
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0answers
35 views

a real analysis question,I need to prove whether two sequences are equidistributed or not,really need some help!

For each subset $S\subset [0,1]$ write $X_{s}$ for the "indicator function" as $X_{s}(t)=1$ when $t\in S$ and $X_{s}(t)=0 $ when $t\notin S$. a sequence $\{x_{n}\}$ in [0,1] is called ...
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1answer
60 views

Use Sigma Formulas and Find the Value of the Sum: $\sum_{i=1}^{n}(7+6i)^2 $ [on hold]

I have $294n+252n^2+252n+2n^3+n^2+2n^2+\frac n 6$ The number seems too large.
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0answers
10 views

Approximating an Appell series with a second-order polynomial

Taking this Appell F1 hypergeometric series $$ f(t)=F_1\left(2;\frac{3}{2} (1-m),\frac{3}{2} (\lceil r\rceil +m-1);3;\frac{\frac{r}{2} t^2 \left(1-\frac{r}{\lfloor r\rfloor +m}\right)+\lfloor r\rfloor ...
2
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4answers
38 views

How can I find if $\sum_{n=2}^\infty {(-1)^n*4 \over (ln(n))^2} $ converges or diverges using the alternating series test?

$$4\sum_{n=2}^\infty {(-1)^n \over (\ln(n))^2} $$ If $4 \over (\ln(n))^2$ = $u_n$, then $u_n$> 0, and: $$\lim_{n\to\infty} u_n = 0 $$ But there is one more test to prove convergence which says ...
0
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1answer
54 views

How can I solve like this exercise

Let we have the following initial value problem : $$y'=f(x,y)=e^y$$ With the condition $y(0)=0$ Find the largest interval $|x| \le a $ makes the initial value problem has an unique solution ...
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0answers
21 views

limit of recurrence serie by derive

how to define limit this series $$U_{n+1} = {df(U_n)\over dU_n}$$ $f(x)$ function derivable in Class $C^n$, $f^{(n)}(x)$ function not null and $U_0 \gt 0$
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0answers
21 views

True/false statements about convergence of a sequence

Let $(a_n)$ be a sequence. True or false: If $\lim_{n\to\infty}(a_{2n}-a_n)=0$ then $(a_n)$ converges. If $(a_n)$ converges then $\lim_{n\to\infty}(a_{2n}-a_n)=0$ Intuitively, I ...
0
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1answer
55 views

How simplify this sum?

I need help to simplify this sum : $$\sum_{i=0}^{x-1}\left(1-\dfrac{1}{2^i}\right)^{m-1}$$ Is it possible to simplify it ? Thank you.
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0answers
46 views

When can we assume this identity about series?

Suppose we are given that $$\sum_{i =a} ^b f (i)g (i) =\sum_{i =a} ^b f (i)h (i) \tag {1} $$ when can we conclude that $$\forall i =a,…,b:g(i)=h(i) \tag {2} $$ ? For example, if $g, h $ are both ...
2
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1answer
55 views

How to evaluate $\sum\limits_{n=1}^\infty (-1)^{n-1} \ln (1+\frac1n)$

Can someone help me evaluate the sum of this series through elementary means? $$\sum_{n\geq 1}(-1)^{n-1} \ln \left(1+\frac1n\right)$$
2
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4answers
92 views

Can I find $\sum_{k=1}^n\frac{1}{k(k+1)}$ without using telescoping series method?

I am interested to find $$\sum_{k=1}^n\frac{1}{k(k+1)}$$ without using telescoping series method. I tried very hard but still could not think of a way to find it without using telescoping series ...
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4answers
55 views

Root test for $\sum_{n=8}^\infty \left(1+\frac{3}{n}\right)^{n^2}$

I got that it would be infinity and diverge, but my answer seems to be incorrect. What did I do wrong? $$\lim_{n\to\infty}\left(1+\frac3n\right)^{n^2}= \lim_{n\to\infty} ...
1
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1answer
60 views

evaluate trigonometric

To all the genius out there , here is a question about expresssing summation of hyperboilc functions : First of all, I've already proved that: $$\sinh(x + 1)- \sinh(x) = (-􀀀1 + \cosh (1)) \sinh(x) ...
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0answers
34 views

Calculating Infinite Sums [on hold]

How do you calculate sum of infinitely sequenced numbers in an arithmetic or geometric progression? Please help me out! Thanks in advance!
0
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2answers
34 views

Could someone help me clarify the steps for this solution?

Given $$\sum_{n=1}^\infty \frac{1}{n^6} = \frac{\pi^6}{945},$$ calcuate $$\sum_{n=1}^\infty\frac{1}{(n+2)^6}.$$ Solution: $$\sum_{n=1}^\infty\frac{1}{(n+2)^6} = \frac{1}{3^6} + \frac{1}{4^6} + ...
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0answers
34 views

Series verification

Anyone can tell me what series is this? As I heard that this kind of series already been long understood. I am required to calculate the value of $P_2$ from the 1st sequence, the value of $P_2$ is ...
4
votes
4answers
63 views

The sequence $\frac{2}{2-u_n}$ diverges

Let $(u_n)$ be a sequence defined with $u_{0}$ a real number such that $u_0 \notin \{0,1,2\}$ and $$u_{n+1} = \frac{2}{2-u_n}$$ Prove that $(u_n)$ diverges. I try to use the fact that this sequence ...
2
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1answer
35 views

convergent series and divergent series

Hi I have two questions. First, $\sum_{n=1}^\infty \frac{n}{n^3+1}$. Is it divergent or convergent? I think it seems like it is positive and decreasing function so we can apply integral test. ...
3
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2answers
39 views

Multiple part problem concerning the proof that $\sum_{k=1}^n k^3=\left(\frac{n(n+1)}{2}\right)^2$ by induction

So I'm having trouble with $c,d$ and $e$. For $c$ so far I have: Inductive Hypothesis: $(\frac{n(n+1)}{2})^2 = (\frac{(k+1)(k+2)}{2})^2$ is that correct?
3
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2answers
59 views

How can I find the convergence/divergence of $\sum_{n=1}^\infty {n! \over n^n}$

Using either root test or ratio test. I have the feeling that it is the root test, I'm not sure how to proceed from this: $$ \sqrt[n]{n! \over n^n}= {(n!)^{1\over n} \over n} $$
1
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1answer
23 views

Derive an exact formula (solve the recurrence definition) for the following recursive sequence:

Derive an exact formula (solve the recurrence definition) for the following recursive sequence: $s_n = 2_{s_n-1} - s_{n-2}$ where $n \ge 2$, and $s_0 = 4$, $s_1 = 1$. So I saw someone solving this by ...
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3answers
68 views

How can I find if $\sum_{n=1}^\infty {n! \over 10^n} $ converges or diverges?

$$\sum_{n=1}^\infty {n! \over 10^n} $$ I wasn't sure on which method to use, I think the ratio test might work, but I'm stuck. Here's what I have so far: $a_n$= $n! \over 10^n$ & ...
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1answer
38 views

Proof that $\lim \frac{a_n}{1+a_n^2} = 0 \implies \lim a_n = 0$

I´ve tried some exercises about sequences convergence, particularly: Let $a_{n}$ be a sequence such as $\displaystyle\lim_{n\rightarrow\infty}\frac{a_{n}}{1+a_{n}^2}=0.$ Prove that $a_{n}$ ...
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2answers
34 views

Prove that every natural number is a limit of a subsequence

I need to prove that every natural number is a partial limit (limit of a subsequence) to the sequence ${{a}_{n}}=n-{{\left\lfloor \sqrt{n} \right\rfloor }^{2}}$ . I already found that 0 is a partial ...
2
votes
2answers
49 views

How to find convergence/divergence of this series

$$\sum_{n=1}^\infty {1+\cos(n) \over n^2}$$ I used the comparison test and said that $\sum_{n-1}^\infty {1 \over n^2}$ is comparable and also larger than $\sum_{n=1}^\infty {1+\cos(n) \over n^2}$, ...
2
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1answer
75 views

Showing that $\sum\limits_{n=1}^{\infty} (a_1+2a_2+…+na_n)/n(n+1) = \sum\limits_{i=1}^n a_n $

Let $\sum\limits_{n=1}^{\infty} a_n$ a series of positive terms convergent. Show that $\sum\limits_{n=1}^{\infty} \frac{a_1+2a_2+...+na_n}{n(n+1)}$ converges to the same value of $\sum\limits_{i=1}^n ...
1
vote
0answers
26 views

How would I determine if this infinite series is convergent or divergent using the limit comparison test?

$$\sum_{n=1}^\infty {2^n \over3+4^n}$$ My thinking is that $4^n$ will grow much more rapidly than $2^n$, and the +3 in the denominator is negligable. Therfore, I should compare it to ...
5
votes
1answer
61 views

Solving $y=\prod_{n=1}^{\infty}\frac{d^ny}{dx^n}$

There is the trivial $y=0$, but beyond that, could there be further solutions for $y$ in terms of $x$ such that $$y=\prod_{n=1}^{\infty}\frac{d^ny}{dx^n}\mbox{ pointwise}$$ ? I posed this problem ...
0
votes
1answer
29 views

estimates for Abel's theorem

Suppose $a_1,a_2,\dots$ is a sequence of real numbers with $\displaystyle\sum_{n=1}^\infty a_n =s<\infty$. For $0<z<1$, define $f(z):=\displaystyle\sum_{n=1}^\infty z^n a_n$. By Abel's ...
0
votes
1answer
57 views

If $(y_{2n}-y_n) \to 0$ then $\lim_{n\to \infty} y_n$ exists

Assume $$\lim_{n\to \infty} (y_{2n} - y_n)=0$$ then $$\lim_{n\to \infty} y_n$$ exists. I know it's not true, and I can see a sequence that disprove that $(1,1,2,1,3,2,4,1,5,3,...)$ but I want a ...
1
vote
1answer
54 views

Infinite series $\sum_{n=1}^{\infty}nx^{n+1}$ does not comply to any of my (known) tests

I am attempting to find the interval of convergence for $$\sum_{n=1}^{\infty}nx^{n+1}$$ The lower bound, x = -1, would be tested by determining if $$\sum_{n=1}^{\infty}n(-1)^{n+1}$$ diverges. ...
0
votes
0answers
27 views

Given $f=x^4+x+1 \in \mathbb Z_{2}[x]$ is primitive, write down an $m$-sequence ${a_n}$ associated to $f$

I'm not sure how to solve this question exactly. I know that the period will be 15 but I don't know how to construct the $m$-sequence.
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0answers
24 views

Geometric and arithmetic sequence [on hold]

The fourth term of a geometric sequence is -64.The product of the first and third terms is 16.Which of the following statements is/are true? A.The first term and common ratio have opposite signs B.The ...
4
votes
1answer
50 views

Find a series convergent $\sum a_n$ such that $\sum\sqrt{a_n/n}$ diverges.

This is part of an exercise 8.22 from Apostol's Mathematical Analysis. I've looked at things like, $a_n=1/\sqrt n-1/\sqrt{n+1}$, and $a_n=1/\log n^{\log n}$, but I can't seem to find anything that ...