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

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2
votes
1answer
33 views

Divergent succession, but with convergent sum average.

An example of a sequence $a_n$ such that: $$a_n\rightarrow\pm\infty$$ but $$b_n=\frac{\sum_{k=1}^{n}a_k}{n}$$ converge.
4
votes
2answers
56 views

Binomial Theorem of Differentiation? [duplicate]

I noticed that $$\frac{d^{n}}{dx^{n}} f(x)g(x)=\sum_{i=0}^n {n \choose i} f^{(i)}(x)g^{(i)}(x)$$ and it's had me scratching for a little bit. It's easy to see how the cross terms add up but can anyone ...
0
votes
0answers
41 views

An interesting question about sequence.

Let $a^{(k)}=(a_j^{(k)},j=1,2,3,...) \in l_{\infty}$ be a sequence such that $\|a^{(k)}\|\le M$ for all $k-1,2,3,...$. Show that there exists a sub-sequence $a^{k_m}$ and $a\in l_\infty$ such that ...
2
votes
2answers
27 views

Series Radius/interval of convergence

Help please! I have no idea how to do this question. I tried using the ratio test
1
vote
5answers
31 views

Infinite sequence and power series

infinite sequence $a_{n}$ where $$\lim_{n\to \infty} |na_{n}|=1101 $$ Find R of convergence of the power series $$\sum_{n=1}^\infty a_{n}x^n$$ Anyone can guide me for this question? Thank you so ...
6
votes
2answers
76 views

Exploring $ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$.

I was exploring the fact that $$ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe,$$ where $B_n$ is the $n$th Bell number. I found this result by exploring the series on wolframalpha and looking up the ...
1
vote
1answer
27 views

Ratio test cancellation trouble

$$\sum\limits_{n=8}^{\infty}\frac{6^n}{(2n)!} $$ Can someone walk me through the cancellation of numbers in this ratio test problem? I seem to be forgetting something and its leading me to the ...
2
votes
1answer
37 views

How to prove the limit of “the exponential of a sequence”

So given a convergent sequence $\{a_n\}_{n=1}^\infty$ with limit $a$, I'd like to prove that $$\lim_{n\to\infty} \left(1+\frac{a_n}{n}\right)^n=e^a.\quad(1)$$ Knowing that $e$ is defined by ...
0
votes
1answer
32 views

Ratio test cancellation when applied to $\sum n 7^n/(n+2)!$

I am having trouble canceling out numbers using the ratio test. I got the denominator correct but I don't see how the numerator is not 7.
3
votes
3answers
73 views

Does $\sum_{n=1}^\infty \frac{\cos(n\pi/3)}{n!}$ absolutely converge?

Using the Ratio Test, I have to find whether $$ \sum_{n=1}^\infty \frac{\cos(n\pi/3)}{n!} $$ converges or diverges. The back of the book says that the sum is absolutely convergent. My work: $a_n = ...
0
votes
4answers
47 views

Find limit of $x_n =n2^{-n}, n\in\mathbb{N}$

By writing out the first few terms of the sequence, I see that it is a decreasing sequence (monotonic non-increasing) and want to show it converges to 0. But I don't know how to manipulate the ...
0
votes
0answers
32 views

Prove locally uniformly convergence of a sequence

Let for $n = 0,1,2,...$ , $f_n : [0,1] \rightarrow \mathbb{R}$ defined by $f_n (x) = x^n$. 1) Is the convergence of {$f_n$}$_{n=0} ^\infty$ to $f$ locally uniformly on the interval $[0.1]$? 2) And ...
1
vote
4answers
61 views

Find the limit of a sequence defined by $x_n =\sqrt{n^2+1}-n, n\in\mathbb{N}$

I want to use the standard definition $x_n \rightarrow x$ if for all $\epsilon>0$ there exists $N$ such that if $n>N$ then $|x_N-x|<\epsilon$. So my claim is $x_n\rightarrow 0$ If I set ...
0
votes
1answer
59 views

Proving that if the sequence $X_n$ converges to $x$, then ${X_n}^a$, where $a$ is a positive rational, converges to $x^a$.

I've been stuck on this problem for a while. I splitted a into $p/q$, so it would be $({X_n}^p)^{1/q}$, and I got the convergence of ${X_n}^p$ to be $x^p$ since it is just induction using the product ...
-2
votes
1answer
33 views

Show that $\sum_{n = 1}^\infty n^qx^n$ is absolutely convergent, and that $\lim_{n \rightarrow \infty}$ $n^qx^n = 0$

I'm having trouble with proving the following for my math study: Let $x$ be a real number with $|x| < 1$, and $q$ be a real number. Show that the series $\sum_{n = 1}^\infty n^qx^n$ is absolutely ...
0
votes
2answers
117 views

Calculating two convergent series [closed]

I got stuck calculating the following two limits. I couldn't think of a way of how to begin estimating these series. $\lim_{n\rightarrow \infty} \sum _{k=0}^n (-1)^k \frac {(n-k)!} {k!n!}$ ...
2
votes
6answers
95 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
77 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
4answers
79 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
votes
4answers
54 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
1answer
39 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
vote
1answer
25 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
votes
1answer
38 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 ...
0
votes
2answers
31 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
votes
2answers
29 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
votes
0answers
33 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
vote
2answers
109 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 ...
-4
votes
0answers
24 views

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

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!) + ...
2
votes
1answer
22 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
votes
0answers
34 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
39 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
votes
1answer
34 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
votes
1answer
78 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
votes
1answer
70 views

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

$f(x)$ 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 ...
5
votes
0answers
62 views
+50

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
36 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
46 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. [closed]

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
vote
1answer
42 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
80 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.
1
vote
1answer
64 views

I need to prove whether two sequences are equidistributed or not

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 ...
-2
votes
1answer
62 views

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

I have $294n+252n^2+252n+2n^3+n^2+2n^2+\frac n 6$ The number seems too large.
0
votes
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
39 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
votes
1answer
56 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 $U_0 \gt 0$
0
votes
0answers
25 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
votes
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
votes
0answers
59 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
votes
1answer
61 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? $$\sum_{n\geq 1}(-1)^{n-1} \ln \left(1+\frac1n\right)$$