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

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2answers
41 views

Proof: Convergence of a series implies convergence of another series

Question: $$\sum_{n=1}^\infty a_n $$ is convergent with $a_n$ positive,prove the series $$\sum_{n=1}^\infty \frac {{(a_n)}^{1/2}}{n}$$ is convergent. The hint given is: $x^2+y^2\geq2xy$. But I ...
0
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1answer
14 views

How to show $[0, \frac{\pi}{2}) \cup (\frac{3\pi}{4}, 2\pi)$ is sequentially separated?

Definition of separated: $E$ is separated if $A, B \neq \varnothing$, $A \cup B = E$, and there are not convergent sequences in $A$ that have limit points in $B$, and vice versa. Definition of ...
6
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3answers
107 views

Find the sum of the n first series numbers: $7,77, 777,…$

Find the sum of the $n$ first numbers: $7,77, 777,...$ I thought to find an order by dividing $77/7=11, 777/7=111...$ but I don't know how to continue.
4
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3answers
104 views

Which (convergent) series can one find the sum of?

I know about geometric series and how one can find the sum when they are convergent. I also have heard that one can prove that the $p$-series $$\sum_{n=1}^{\infty} \frac{1}{n^2}$$ has sum $\pi^2/6$ ...
1
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0answers
63 views

Does a Banach valued Cauchy series over arbitrary set converges?

Definitions: If $f$ is a function from a set $A$ (not necessarily countable) into a Banach space $V$, we say that the series of $f$ over $A$ converges if there exists an element $v \in V$ such that ...
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0answers
31 views

Two statements about partial limits and intervals

Let $(a_n)$ be a sequence and $I=(a,b)$ an open interval such that every $L\in(a,b)$ is a partial limit of $(a_n)$. Decide whether or not the following statements are true: $\{a_n | n \in ...
1
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5answers
66 views

How can I compute $\sum_{n=0}^{\infty} 0.6^n$? [duplicate]

I am a computing teacher and just helping out some students with a math question. They have been asked to calculate the following: $$\sum_{n=0}^{\infty} 0.6^n$$ I am intrigued as to how one gets to ...
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9answers
114 views

If $a_n = \frac{e^{n}}{e^{2n}-1}$ how do I show that $a_{n+1} \leq a_n$?

Let $$a_n = \frac{e^{n}}{e^{2n}-1}$$ How do I show that $a_{n+1} \leq a_n$? I don't know how to deal with the $-1$ in the denominator.
0
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0answers
34 views

Which sum formula should I use for arithmetic sequences?

I was looking online for a formula that would allow me to find the sum of arithmetic sequences and I came across the 2 that I have listed below. I tried to use both them and they worked. So, I ...
0
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1answer
30 views

Recursive Sequence with different conditions

I'd like to deduce a formula for a slightly wierd recursive sequence i've got $$ f(n) = \begin{cases} f(n - 1 ) + 1, & \text{if $n$ is even} \\ 2f(n - 1), & \text{if $n$ is odd} \end{cases} ...
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1answer
41 views

If a sequence is decreasing to zero, why do we have the following

Let $\{a_i\}_{i\in\mathbb{Z}}$ be some real sequence and let $S_n = \frac{1}{n}\sum_{|i|<n}^n|a_i|$. I need to verify two claims. First, if $a_n\to0$ as $n\to\infty$, why ...
1
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3answers
43 views

Question Sequences and Series

Prove that the sequence $(a_n)_{n \geq 1}$ defined by $$ a_n= \sum_{k=1}^{n} \frac{1}{k} - \ln(n+1)$$ is increasing and bounded. It is on the study guide for my final exam, which is tomorrow so I am ...
1
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2answers
37 views

Limits and sequences question

Let $$a_n=n^x(n^{1/n^2}−1).$$ Show that $$\lim_{n \to \infty} \frac{a_n}{\ln(n)/(n^{2-x})} = 1. $$ It is on the study guide for my final exam, which is tomorrow so I am trying to figure it out. ...
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0answers
21 views

Series and sequences- Help [on hold]

Let $a_n=n^x(n^(1/n^2)-1)$ for n in natural numbers and assume that lim(n goes to infinity)ln(n)/n^r = 0 for any r>0. Let ln(x)=integral(from t=1 to x)dt/t for x>0. Prove the inequality h/1+h < ...
3
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3answers
57 views

Series Proof $\sum_{k=1}^n (1/k) > \ln(n+1)$

Prove that $\sum_{k=1}^n (1/k) > \ln(n+1)$. I have been trying to do this for some time now, but I cannot figure it out. It is on the study guide for my final exam, which is tomorrow so I am trying ...
4
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3answers
59 views

Proof $\text{Si}(n) $ is convergent

I am trying to prove that the sequence formed by the Si function, $\text{Si}(n) = \int_0^n \frac{\sin(u)}{u} \mathrm{d}u$, is convergent as $n\rightarrow \infty$. The only twist is the lower bound of ...
2
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0answers
13 views

Determining the sequence that yields a balanced search tree in the form of a recurrence / sequence

Let's say I have a sequence of (distinct) monotonically increasing numbers S. I'll want to add them sequentially to a Binary Search Tree (BST) but as the numbers ...
2
votes
0answers
46 views

Series for $\sin(z) / \sin(\pi z)$

${\sin(z) \over \sin(\pi z)} = 1/\pi + {z \over \pi} \sum_{n \in {\bf Z} \setminus \{0\}} {(-1)^n \sin(n) \over n(z-n)}$ First I apply Mittag-Leffler's theorem, to see that the RHS is a mereomorphic ...
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3answers
44 views

If the limit of the sequence exists, find it. If not, prove that the limit does not exist. [on hold]

Consider the following sequence: $\{[\sqrt{n}][\sqrt{n + 1}-\sqrt{n}]\}$ for $ n \geq 1$. If the limit exists, find it and prove that the limit is indeed your choice. If not, prove that the limit ...
2
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2answers
110 views

How to show that $\pi =3+\sum_{n=0}^{\infty }\frac{(2n+1)!!}{(2n+3)(2n+4)(n+1)!2^n}$ [on hold]

How to show that $$\pi =3+\sum_{n=0}^{\infty }\frac{(2n+1)!!}{(2n+3)(2n+4)(n+1)!2^n}$$ I don't have an idea how to start.Any help to prove it?
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0answers
40 views

How many strings $s^\infty$ with $s$ a string of length $\le k$ on alphabet $\{1,2,…,m\}$?

As a function of $k$ and $m$, say $f(k,m)$, how many strings are of the form $sss... = s^\infty$, where $s$ is a string of finite length $\le k$ on the finite alphabet $\{1,2,...,m\}$? E.g., ...
2
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1answer
16 views

Weak continuous convergence of operators

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$. Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$. Let $x_n \rightharpoonup x$ in ...
1
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1answer
20 views

Prob. 6, Sec. 21 in Munkres' TOPOLOGY, 2nd ed: How to show directly that this sequence of functions does not converge uniformly?

For each $n = 1, 2, 3, \ldots$, let $f_n \colon [0,1] \to \mathbb{R}$ be defined by $$f_n(x) \colon= x^n \ \ \ \mbox{ for all } \ x \in [0,1].$$ Then $$ \lim_{n \to \infty} f_n(x) = \begin{cases} ...
1
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2answers
49 views

How can I find the sum of the series $\sum_{n=0}^\infty {(-1)^n \over 4^n}$ or show that it diverges using the geometric series test?

First, I reindexed it: $$\sum_{n=0}^\infty {(-1)^n \over 4^n} = \sum_{n=1}^\infty {(-1)^{n-1} \over 4^{n-1}} = \sum_{n=1}^\infty {\left(-1 \over 4\right)}^{n-1} $$ So now I'm pretty sure it's in the ...
1
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2answers
38 views

Transformation of a function into a power series [on hold]

How can I transform the real functions $\frac{1}{1-\sin(x)}$ and $\frac{x}{e^x-1}$ into power series with $x_0=0$?
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0answers
38 views

Weak continuous convergence of operators

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$. Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$. Let $x_n \rightharpoonup x$ in ...
0
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0answers
25 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
48 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
69 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 ...
0
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0answers
32 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
25 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 ...
1
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3answers
61 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
14 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
29 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 ...
0
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2answers
42 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 ...
0
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1answer
24 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
30 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: ...
1
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0answers
43 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} ...
2
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2answers
33 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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1answer
33 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
35 views

Need help using ratio test [closed]

Only using the ratio test determine where the series converges. $$\sum_{n=1}^\infty \frac{8n!}{n^n}$$
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0answers
39 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
53 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 ...
1
vote
1answer
24 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
votes
1answer
32 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
35 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 ...
6
votes
6answers
955 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 ...
0
votes
2answers
40 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 ...
0
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0answers
16 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)} } ...
1
vote
0answers
27 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 ...