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

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0answers
13 views

Symbol for exponentiation of a sequence? (Equivalent to SIGMA for summation and PI for product)

I have a student asking whether there is a symbol for exponentiation of a sequence? So there's SIGMA for summation of a sequence, PI for multiplication of a sequence and perhaps something else for ...
1
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2answers
41 views

How to show this infinite sum converges uniformly?

Let $f_k$ be a real numbers such that $\sum_{k=1}^\infty f_k < \infty$. For each $R > 0$, define the convergent sum $$v(R) = \sum_{k=1}^\infty f_k(b_k(R)e^{-ky} - c_k(R)e^{ky})$$ where $0 \leq y ...
8
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1answer
65 views

If $\lim_{n\to\infty} (a_{n+1}-a_n)=0$ and $|a_{n+2}-a_n|<\frac{1}{2^n}$ then $(a_n)$ converges

Let $(a_n)$ be a sequence such that $\lim_{n\to\infty} (a_{n+1}-a_n)=0$ and $|a_{n+2}-a_n|<\frac{1}{2^n}$ for all $n$. I have to decide whether or not $(a_n)$ converges. My attempt: I think it ...
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1answer
20 views

Is the sequence $(v_p(n))$ of $p$-adic valuations of positive integers the fixed point of a morphism, for every prime $p$?

Fix a prime number $p$ and consider the sequence $\mathbf{v}_p = (v_p(n))_{n \geq 1}$, where $v_p$ is the usual $p$-adic valuation, i.e. $v_p(n) = a$ iff $p^a \parallel n$. While browsing the OEIS I ...
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0answers
24 views

Shifting limits in series solutions to ODEs

I'm trying to practice the Frobenius method of solving ODEs, and I keep getting the answer wrong. It seems to be down to the shifting of limits of the sums, although it is not clear in the solutions I ...
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3answers
53 views

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

Note: $x_n$ is a sequence which is not necessarily convergent. The following was my attempt. Since $\lim_{n\to \infty}a_n=a$ then $\limsup_{n\to \infty}a_n=a$ . Also ...
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2answers
9 views

Confused about using alternating test, ratio test, and root test (please help).

So I have to determine if $\sum_2^{\infty} \frac{(-1)^n}{ln(n)}$ absolutely converges, conditionally converges, or diverges. So first I tried the Alternating Series Test, because that is what you do ...
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1answer
21 views

Power Series: $\sum_1^{\infty} (x)^{n}\frac{n^3}{n!}$

I just started learning about the power series, can someone help me with finding the radius of convergence and interval of convergence? So I am stuck on the radius of convergence because apparently I ...
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2answers
34 views

Determine if the sequence- $a_1 = 2$ , $a_{n+1} = 72/(1+a_n)$ is convergent.

I've tried to prove that the sequence is convergent by using the monotonic sequence theorem but after computing the first few terms, I realized that the sequence is not monotonic thus, it isn't ...
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1answer
22 views

Determine if the sequence an = ln(n)/(n^(1/n)) is convergent.

I have some difficulty with showing that the sequence $$ a_n = \frac{\ln(n)}{n^{1/n}} $$ is divergent. Can anyone help me out with this? Thanks!
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0answers
27 views

Continuity of exponentiation (Terence Tao's book-Lemma 6.7.1)

How to use the sequence $(x^{1/k})_{k=1}$ convergent at 1 (Cauchy sequence) for testing whether $eventually\ $ $\epsilon\cdot x^{-M}-close$ to 1 ? How to get the next result: ...
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0answers
15 views

The $L^p$ convergence rate of the tail of the series $\sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} \}2^{-na}$

This a follow-up to the question: Convergence Rate of the Tail of the Series $m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-ja}$ When $a > 0$, we have $$ \sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} ...
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2answers
25 views

Common terms between two arithmetic series

There are two arithmetic series. There may be common terms between two sequences. We have to prove whether or not common terms between two series also form an arithmetic series. If yes what is first ...
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0answers
12 views

Upper and lower bounds for functional series

Suppose $x\in[0,a]$, $a>1$. Let $g_0(x)\equiv x$, $g_1(x)=(1+x)/2$, and $g_{n+1}(x)=g_1(g_n(x))=g_n(g_1(x))$. Consider $\{\zeta_i(x)\}_{i\ge0}$ where $\zeta_i(x)$ is defined on the interval ...
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1answer
28 views

Simplified summation formula.

Suppose I have the following recursive formula: $$A(n)=-\sum_{k=0}^{n-1}\binom{n}{k}\frac{1+(-1)^{n-k}+2(n-k)(-1)^{n-k-1}}{2}A(k)$$ Then I can combine the negatives to get ...
0
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1answer
33 views

Need help using ratio test [on hold]

Only using the ratio test determine where the series converges. $$\sum_{n=1}^\infty \frac{8n!}{n^n}$$
0
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0answers
33 views

intuition about calculating partial sums of series

The partial sums $$1 + 2 + 3 + \cdots + n$$ of the simple arithmetic progression can be calculated by reordering and adding. The partial sums $$1 + \frac{1}{2} + \frac{1}{4} + \cdots + ...
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1answer
50 views

Playing around with Geometric series. Where is my algebraic mistake?

Playing around with geometric series I got confused. First, consider the following equation $$\sum_{i=0}^n 1^i = n+1$$ Now, replacing $1$ by $\frac{a}{a}$ gives $$\sum_{i=0}^n ...
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1answer
23 views

Asymptotic behaviour of sequences

Could anyone explain in details how these approximations as $n \to \infty$ are found? ($a$ is a positive real number) ${x_n} = \frac{1}{n}\left( {\frac{a}{3} - \frac{3}{2}} \right) + O\left( ...
0
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1answer
30 views

How to notate the final element in a sequence?

I'm having troubles putting this in to words here, but here it goes: If I have a sequence of numbers, called $A$ where $A$ is a sequence of numbers that don't seem to have a pattern, how can I notate ...
1
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0answers
29 views

What causes long sequences of consecutive 'collatz' paths to share the same length?

I asked Longest known sequence of identical consecutive Collatz sequence lengths? some time ago, but I don't feel like it really got to the bottom of things. See, in the answers lopsy find a sequence ...
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6answers
915 views

If each term in a sum converges, does the infinite sum converge too?

Let $S(x) = \sum_{n=1}^\infty s_n(x)$ where the real valued terms satisfy $s_n(x) \to s_n$ as $x \to \infty$ for each $n$. Suppose that $S=\sum_{n=1}^\infty s_n< \infty$. Does it follow that ...
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2answers
35 views

How to fastest approximate definite integrals

I know that a definite integral is a limit of Riemann sums. So if one wanted to estimate a definite integral (because one might not be able to find an antiderivative), then one can just take enough ...
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0answers
15 views

Series representation of simple function - a general form for the coefficients?

I'm looking for a series representation for $$ f\left(r_j\right)=\frac{ \left( m - r_j \right)^{\frac{3}{2}\left(m-1\right)}}{\left(j + m - r_i - r_j \right)^{\frac{3}{2} \left( m + j - 1 \right)} } ...
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0answers
24 views

Convergence Rate of the Tail of the Series $m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-na}$

When $a > 0$, it's fairly easy to see that $$ m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-na} \leq C \quad \forall m \in \mathbb{N}, $$ where $C$ is a finite constant independent of $m$. How ...
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1answer
39 views

Find the last 2 digits of the 200 digited number 123012300123000…

Here i've tried a method..Though $[123012300123000\dots]$ is a number, but for sake of solving the sum I've separated into some terms.. something like this- ...
2
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1answer
32 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.
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2answers
52 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
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0answers
34 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
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2answers
25 views

Series Radius/interval of convergence

Help please! I have no idea how to do this question. I tried using the ratio test
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5answers
28 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
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2answers
68 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
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1answer
22 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
34 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
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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
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3answers
70 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
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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
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0answers
28 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 ...
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4answers
60 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
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1answer
54 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 ...
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1answer
31 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 ...
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2answers
114 views

Calculating two convergent series [on hold]

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
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6answers
91 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
74 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
77 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
52 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
33 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
23 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
37 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
30 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 ...