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

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1
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1answer
42 views

Show that {$a_n$} is convergent and find sup{$a_n| n \in Z_+ $}

$a_1 = 1$ and $ a_{n+1} = \frac{4+3a_n}{3+2a_n} ; \forall n \in Z_+$ Show that {$a_n$} is convergent, find its limit and find sup{$a_n| n \in Z_+ $} if exists. I found the limit as follows - ...
3
votes
2answers
26 views

radius of convergence when root test fails

I'm stuck on this problem: Find the radius of convergence of $\sum \limits_{n=1}^\infty \frac{x^n}{((2+(-1)^n)^n} $ An attempt: From the root test, it seems $L$ does not exist: $$L= \lim_{n ...
1
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2answers
48 views

Find a closed form for the generating function for this sequence

The sequence: $0, 0, 0, 1, 1, 1, 1, 1, 1, \ldots$ The book gives the answer of $\frac{x^3}{1-x}$ but I'm not sure how to get this answer. I understand the generating function of this sequence will be ...
1
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2answers
44 views

A trouble with the discrete product topology

Consider a finite set $S=\{1,\ldots,n\}$ with the discrete topology, and moreover construct the product topological space $S^\mathbb N$ with the product topology. $S^\mathbb N$ is made by all the ...
3
votes
2answers
52 views

Infinite series involving sum of divisors function

Is there much known about infinite series of the form $\sum_{n=1}^{\infty}\frac{\sigma_{1}(n)}{n}q^{n}$ where $\sigma_{1}(n)$ is the sum of divisors function. I am particularly interested in ...
0
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0answers
32 views

Convergence of a series using limit test

As per my understanding: When we take $\lim_{n \rightarrow \infty}$ of a function, it should approach a finite number, it converge. And if the opposite is true, it diverges. Now, the test for ...
0
votes
1answer
19 views

Mixing arithmetic and geometric progressions

I'm having trouble blending two different types of progressions: The fourth, eighth and fourteenth terms of an A.P., common difference 0.5, are in geometric progression. Find the first term of the ...
5
votes
2answers
120 views

From the series $\sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)$ to $\zeta(\frac{1}{2}+it)$

Here is a pretty series $$ \displaystyle \sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)=\frac{1}{2} \left(1-\ln (2\pi)+\gamma\right) \quad (*) $$ where $H_{n}:=\sum_{1}^{n} ...
4
votes
6answers
198 views

Find the $n$th term of $1, 2, 5, 10, 13, 26, 29, …$

How would you find the $n$th term of a sequence like this? $1, 2, 5, 10, 13, 26, 29, ...$ I see the sequence has a group of three terms it repeats: Double first term to get second term, add three to ...
0
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0answers
47 views

Another proof of the Dirichlet's test

My teacher said, that the Dirichlet's test was equivalent to the lemma as follows, and the lemma could be proved with an estimate without using Abel's summation formula. He expected me to complete the ...
0
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1answer
20 views

Geometric Progressions: Finding the number of terms that will double the first term

If the value of an article is assumed to increase annually by 5% of its value at the beginning of the year, after how many years will its value double. Here is what I've done so far: Value at ...
0
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1answer
6 views

Finding Possible Values of GP Common ratio (r)

r is the common ration of a GP (r is not equal to 1) and the sum of the first 4 terms is 5 times the sum of the first 2 terms. Find the possible values of r. How do I solve this one? Thanks.
2
votes
2answers
39 views

Integral to measure error within 10^-8

If someone could give me background on HOW to solve this problem, NOT THE ANSWER, that would be appreciated. I would love to know how to approach this problem in the most efficient and universal way. ...
4
votes
2answers
68 views

An asymptotic term for a finite sum involving Stirling numbers

The question is a by-product at the end of this post. The following asymptotic term will ensure the convergence of some series. $$ \frac{1}{n!} \sum_{k = 1 }^{n } \frac{{n \brack k}}{k+1} = ...
4
votes
3answers
355 views

I believe this is a Taylor series. How do I approach it, and what formulas can I use to solve this type of problem?

Suppose that $|x| < 1$. Find the sum of the series $$2x - 4x^3 + 6x^5 - 8x^7 + \cdots$$ I'm not looking for an answer. I want to know how to appropriately solve such a question though.
1
vote
1answer
28 views

Property of Conditionally Convergent series

If $ \sum a_n$ be an conditionally convergent series.For any real number R, is it true that there exists a sequence$\{b_n\}$ where each $b_i=1 $ or $-1$ such that $\sum a_nb_n$ converges to R?
4
votes
3answers
110 views

Comparison test for sequences?

Let $a_n, b_n$ such that for sufficiently large $n$: $ a_n \le b_n$. Can we deduce that: $\lim_{n\to\infty}a_n = \infty \implies \lim_{n\to\infty}b_n = \infty$ $\lim_{n\to\infty}b_n = L ...
0
votes
2answers
123 views

Fibonacci series, which is most pure mathematically? [closed]

There are various methods of generating the Fibonacci series. I'm going to list 3 of them in this question. Method 1: f[n] = f[n - 1] + f[n - 2] Method 2: (more ...
1
vote
2answers
74 views

prove ${a_n} = \root n \of {n!} $ is monotonically increasing to $\infty$

Prove ${a_n} = \root n \of {n!} $ is monotonically increasing to $\infty$ I already showed that $a_n$ diverges to infinity like this: I used to the lemma which says that if ...
0
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2answers
25 views

Associative property for series

Are those equation always valid: $$\sum a_n + b_n = \sum a_n + \sum b_n$$ $$\sum_{k=1}^n(a_{k+1}+a_k)-\sum_{k=1}^na_k=\sum_{k=1}^na_{k+1}$$
0
votes
1answer
34 views

Prove that the elements of these two sequences are not null

Let $x_{n+1}=x_n+2y_n$ and $y_{n+1}=y_n-x_n$, where $x_1=1$ and $y_1=-1$. I tried proving by contradiction, I tried by induction, I got nothing. This is a question I had on an exam, I didn't manage ...
2
votes
1answer
66 views

What's an intuitive way to compute summation of this series?

What's an intuitive way to compute $$\log(1)+\log (2)+\log (3)+\cdots+\log (n-1)+\log (n)$$ or for $n>a$ $$\log(a)+\log (a+1)+\log (a+2)+\cdots+\log (n-1)+\log (n) $$ I know the answer for ...
3
votes
1answer
45 views

investigate $\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$

I need to investigate the series (Hence, when the series converges and when the series converges absolutely depending on $\alpha$). $$\sum\limits_{n\ge2}{\frac{(-1)^n}{n^\alpha \ln n}}$$ For ...
4
votes
1answer
114 views

I don't understand the solution to this limit.

$\lim _{n\to \infty }\left(\frac{1+5+5^2+...+5^{n-1}}{1-25^n}\right)$ I have a solution to this question, but I don't really understand it. It's: $\lim _{n\to \infty \:}\left(\frac{1+5+5^2+\ldots ...
2
votes
1answer
38 views

$\lim_{n \rightarrow \infty} \frac{3}{n} \sum_{k=1}^n \left(\frac{2n+3k}{n} \right)^2$

How to solve this: $$\lim_{n \rightarrow \infty} \frac{3}{n} \sum_{k=1}^n \left(\frac{2n+3k}{n} \right)^2$$ The answer is supposed to be 39. My attempt: ...
4
votes
1answer
33 views

A question about the convergence of series.

If$\hspace{0.3cm}$ $\sum a_n$ and $\hspace{0.3cm}$$\sum b_n$$\hspace{0.3cm}$ are convergent then which of the following is true? $1.$$\hspace{0.3cm}$$a_{n+1}<a_n$$\hspace{0.3cm}$ $\forall n$ ...
2
votes
1answer
50 views

Series in a space which is not complete

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
2
votes
0answers
43 views

Number of compositions, does this sequence have a closed form?

Does this sequence have a closed form: 1, 1, 2, 3, 5, 8, 14, 25, 46, 87, 167, 324, 634, 1248, 2466, 4887, 9706, 19308, 38455, 76659, 152925, 305232, 609488, 1217429, 2432399, 4860881, 9715511, ...
3
votes
4answers
484 views

Numbers whose digits sum to 7

Let $S$ be the sequence of all positive integers whose decimal digits add to exactly 7, in increasing order: $$S = \langle7, 16, 25, \ldots, 70, 106, 115, 124, \ldots 160, 205, \ldots, 10230010, ...
1
vote
0answers
23 views

Find limit property

Let $f$ be a function on $\mathbb{R}$ satisfy: $|f(x)-f(y)|\leq|x-y|$ $\forall x,y\in\mathbb{R}$. Consider the sequence: $$u_{n+1}=\frac{u_n+f(u_n)}{2},u_0=a$$ Research the limit property of this ...
3
votes
5answers
107 views

Taylor expansion of $\frac{1}{1+x^{2}}$ at $0.$

I am trying to find the Taylor expansion for the function $$f(x) = \frac{1}{1+x^{2}}$$ at $a=0.$ I have looked up the Taylor expansion and concluded that it would be sufficient to show that ...
2
votes
3answers
57 views

Find the 325th term of the series 7,16,25,34…

One of my friend gave me the series 7,16,25,34,43... I figured it out easily that the sum of digits is 7 in each case. How can I find the 325th term of this series? Also is there any trick/formula to ...
0
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2answers
29 views

Find power series representation of $ x/(x^{2}+9)^{2}$

I'm not sure how to do it since the entire bottom term is squared. Is there a geometric series I should use? Or differentiation?
0
votes
2answers
22 views

Prove for a monotone function, if $x_0$ is interior on interval $I$ then the one-sided limits exists

The proof goes like this: Lets suppose $f$ is nondecreasing (for nonincreasing we'll observe the function $-f(x)$). Let $x_0$ an interior point in the interval $I$, and $\left\{x_k\right\}$ an ...
5
votes
3answers
159 views

Examples of “difficult” integrals with are easier to solve with a series?

Yesterday someone posted an extremely elegant solution to a seemingly bizarre series where the integral: $$\int_{0}^{1} x^{m}\ dx = \frac{1}{m + 1}$$ was utilized. Oftentimes one will also ...
4
votes
1answer
98 views

Is the sequences$\{S_n\}$ convergent? [duplicate]

Let $$S_n=e^{-n}\sum_{k=0}^n\frac{n^k}{k!}$$ Is the sequences$\{S_n\}$ convergent? The following is my answer,but this is not correct. please give some hints. For all $x\in\mathbb{R}$, ...
1
vote
1answer
47 views

sum of the series $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$ [closed]

Let $|x_n|<1$. Is the given series convergent or which conditions do we impose on the sequence $(x_n)$ to given series be convergent? $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$
1
vote
2answers
72 views

Find $R$ such that $\sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(R)}^n\cdot{(1-R)}^{3k-n}$ is constant for all $k\in\mathbb{N}$

Given $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(R)}^n\cdot{(1-R)}^{3k-n}$ with $0<R<1$. The sequence $A_k$ seems to be decreasing for $R\leq0.6$ and increasing for $R\geq0.8$. How can ...
1
vote
1answer
19 views

Is $(\frac{1}{1-|x_n|})$ bounded or only bounded above if $|x_n|<1$? [closed]

Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(\frac{1}{1-|x_n|})$ bounded or only bounded above if $|x_n|<1$?
1
vote
1answer
48 views

Is $(|x_n|^{n+1})$ convergent or bounded if $|x_n|<1$?

Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(|x_n|^{n+1})$ convergent or bounded?
3
votes
2answers
83 views

Prove by induction that $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ is decreasing

I want to prove that the following sequence is monotonously decreasing: $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ I think it should be ...
6
votes
2answers
278 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
13
votes
6answers
2k views

A strange puzzle having two possible solutions

A friend of mine asked me the following question: Suppose you have a basket in which there is a coin. The coin is marked with the number one. At noon less one minute, someone takes the coin ...
3
votes
3answers
99 views

Trouble evaluating the sum involving logarithm

I was trying to solve this problem: Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$ In the procedure I followed, I came across the following sum: ...
3
votes
3answers
74 views

Closed form for the sum: $\sum_{n=1}^{\infty}\frac{1}{n(n + 1/3)}$ [duplicate]

I tried using partial fractions to compute the sum of the series $$ \sum_{n=1}^{\infty}\frac{1}{n(n + 1/3)} $$ Another technique is to turn this series into a definite integral of 0 to 1. but do not ...
7
votes
3answers
304 views

Sum related to zeta function

I was trying to evaluate the following sum: $$\sum_{k=0}^{\infty} \frac{1}{(3k+1)^3}$$ W|A gives a nice closed form but I have zero idea about the steps involved to evaluate the sum. How to approach ...
2
votes
2answers
39 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then find whether the above series ...
0
votes
3answers
49 views

How can I define a “formula” for general term of a sequence with some given values?

I have a doubt: If I have $\alpha, \beta, \gamma, \delta$ natural numbers, how can I write a formula to generate infinite sequences, such that $f(1)=\alpha, f(2)=\beta, f(3)=\gamma, f(4)=\delta$? I ...
-2
votes
3answers
169 views

How many is 1+2+3+… [duplicate]

Let $$S=1+2+3+\cdots=\sum_{n=1}^{\infty}n$$ What is the value of $S$? Some books says that $S=\infty$, other says that $S=-\frac{1}{12}$ and there are some books saying that this is a divergent ...
0
votes
1answer
43 views

How to prove this inequality by induction?

Suppose that $(v_n)$ is a sequence of positive real numbers with $v_1=1$ and such that $$ v_{n+1} \leq v_{n}+ \sqrt{v_{n}^2+1}. $$ How prove that $$ v_{n}\leq 2^n-1 $$ for any integer $n \geq 2$?