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

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1answer
17 views

calculating two convegrent series

I got stuck calculating these two limits. I couldn't think of a way on how to begin estimating these series. $\lim_{n\rightarrow \infty} \sum _{k=0}^n (-1)^k \frac {(n-k)!} {k!n!}$ ...
2
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6answers
51 views

Prove that if $\lim_{n\to\infty}\frac{x_{n+1}}{x_n} = L < 1$ then $\lim_{n\to\infty} x_n = 0$

Let $(x_n)$ be a sequence with $x_n > 0$ for all $n \in \mathbb{N}$. I would like a hint on how to prove that if $\lim_{n\to\infty}\frac{x_{n+1}}{x_n} = L < 1$ then $\lim_{n\to\infty} x_n = 0$. ...
3
votes
1answer
59 views

Find x,y & z (xyz+xyz=zyx)

I saw this problem the other day at work and found it pretty interesting: $$xyz + xyz = zyx$$ Find $x, y, z$ and the base(s) which this is true. Note that $x,y,z$ are simply digits concatenated, ...
3
votes
3answers
57 views

Sum of Square roots formula.

I would like to know if there is formula to calculate sum of series of square roots $\sqrt{1} + \sqrt{2}+\dotsb+ \sqrt{n}$ like the one for the series $1 + 2 +\ldots+ n = \frac{n(n+1)}{2}$. Thanks in ...
0
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4answers
47 views

Solution verification: $\sum_{n=1}^\infty \frac{9^n}{3+10^n}$

I need to find out whether $$\sum_{n=1}^\infty \frac{9^n}{3+10^n}$$ converges or diverges using the limit comparison test. Here's my work: Let $a_n$ be $\frac{9^n}{3+10^n}$, $b_n$ be ...
0
votes
0answers
45 views

If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ . [on hold]

I have tried a lot to solve this problem but am getting nowhere. Could someone please show me how it's done. Thanks. Note: $x_n$ is a sequence which is not necessarily convergent.
0
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0answers
17 views

Finite amount of primes in sequence

I want to prove that there is infinite growing sequence $a_1, a_2, \ldots,$ where for every integer $k \geq 0$, sequence $a_1 + k, a_2 + k, a_3 + k \ldots$ contains only finite amount of primes
1
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1answer
13 views

Almost sure convergence, arithmetic mean, variance

I am stuck proving that this sequence $$\sigma^2_n:= \frac{1}{n-1} \sum_{i=1}^n (X_i - \frac{X_1+...+X_n}{n})^2$$ is converegent almost surely to $D^2X_i = \sigma^2$. We assume that $X_1, X_2, X_3, ...
0
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1answer
30 views

L'hospital's rule for sequences

A similar question appeared here Sequence version of L'Hospital's Rule for example, but something is still unclear for me.. If I only had L'Hositals rule for sequences in lecture and don't ...
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2answers
27 views

Where is the following sequence convergent/absolute convergent?

I have the following sequence: $\sum_{n=1}^\infty x^n\tan \frac{x}{2^n}$ Any idea how to decide this question? It is obvious that $x^n$ goes to infinity if $|x|>1$, but how does the $\tan ...
0
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2answers
28 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
32 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
106 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
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1answer
21 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.
0
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0answers
32 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
votes
3answers
37 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
31 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
66 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 ...
2
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1answer
32 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 ...
0
votes
0answers
15 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 ...
2
votes
1answer
27 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 ...
0
votes
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
votes
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. ...
1
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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
votes
2answers
67 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
37 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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votes
1answer
61 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.
0
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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
votes
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 ...
0
votes
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
56 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.
0
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0answers
53 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
56 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
votes
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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votes
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
63 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) ...
0
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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} + ...
0
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0answers
36 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
votes
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
votes
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
votes
2answers
60 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
vote
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 ...
1
vote
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: $a_n$= $n! \over 10^n$ & ...
1
vote
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 $a_{n}$ ...
1
vote
2answers
35 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 ...