Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

6,952 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
-1
votes
2answers
35 views

How can one show that $(U_n)$ is arithmetic progression

Given that $(U_n)$ a numerical sequence such that : $U_0$=3 $U_{n+1}=\sqrt{(U_n)^2+8n+16}$ Show that $(U_n)$ is a arithmetic progression. So I have to show that $U_{n+1}-U_n=r$ where $r$ is a ...
-1
votes
1answer
255 views

compound interest with monthly deposit

$150 is deposited into an account at beginning of each month that pays 6% compounded monthly. What is the account's value after 10 years? I know how to find the sum using regular compound interest, ...
-1
votes
2answers
100 views

Why $\sum_{n=0}^{\infty}(-n)^3e^{{-n}^{-4}}$ absolutely converges?

I was wondering why in some exercise I found it says $\sum_{n=0}^{\infty}(-n)^3e^{{-n}^{-4}}$ absolutely converges . If it is absolutely convergent then it means that $\sum_{n=0}^{\infty}(n)^3e^{{-...
-1
votes
2answers
96 views

proof of divergence of the serises

Let $\sum c_n$ be a series of positive numbers. Assume that $$\lim_n {{c_{n+1}}\over {c_n}} = r$$ If $0\lt r\lt 1$ then the series converges ; if $r\gt 1$ then diverges . Now , if $r\gt 1$ ...
-1
votes
1answer
98 views

Convergent subsequences common to two bounded sequences

Suppose $( a_n ) _n$ and $( b_n ) _n$ are two sequences of real numbers (not necessarily Cauchy or convergent) Suppose $| a_n | < 2 \ \forall n$ and $| b_n | < 17 \ \forall n$. Prove that there ...
-1
votes
1answer
22 views

Limit is infinite or finite?

The given function is $${{Log\ z}\over {z-1}}=1- {1\over 2}(z-1)+ {1\over 3} (z-1)^2-{1\over 4}(z-1)^3+.... $$ Then it is said that the function tends to $+\infty$ as $z$ tends to $0$ . But ...
-1
votes
1answer
55 views

How to prove that these sequences both converge and have the same limit

Prove that the sequences $\frac{2a_nb_n}{a_n+b_n}$ and $\sqrt{a_nb_n}$ are both convergent to the same limit for all ai,bi>0. Given the special case of two positive numbers $a_i$ and $b_i$, the ...
-1
votes
1answer
26 views

Question about the Convergence of Infinite series due to Cauchy

If there exists $r \in \mathbb{R}$ with $r < 1, K \in \mathbb{N}$, such that $|X_n|^{1/n} < r$ for all $n > K$, then the series $X_n$ (Summation of $X_n$ from $n$ to $\infty$) is absolutely ...
-1
votes
1answer
52 views

How to find value of i when ∑ from k=1 to i is defined by a recursive formula and equals 982?

Thanks for the pointers! Here's updated and edited question I'm trying to find the number of days it takes to reach 982 miles when you start traveling at 18 miles/day and decrease your speed by 2% ...
-1
votes
1answer
81 views

Center of gravity of a sequence

I have a problem to solve that consists in finding a frequency domain expression of this expression, the center of gravity of a sequence. I have tried in several manners but no sucess so far. Does ...
-1
votes
1answer
310 views

Epsilon-N limit Proof for product law for limit

Let the $\lim_{n\to \infty} x_n=a$ and let $\lim_{n\to \infty} y_n=b$. Let $\epsilon>0$ for some $n\ge N$ My notes says: $|(x_n)(y_n)-ab|=|x_n(y_n-b)+b(x_n-a)|$ Can someone show me the ...
-1
votes
1answer
77 views

Decide convergence of the series .

I have problem with these two: a) $\displaystyle \sum_{n=2}^{\infty} \frac{1}{(\ln{\ln{n}})^{\ln{n}}}$ b) $\displaystyle \sum_{n=3}^{\infty} \frac{1}{n \cdot \ln{n} \cdot \ln{\ln{n}}}$ My try: a) ...
-1
votes
1answer
60 views

Evaluating limit of a sequence

Prove: $$\lim_{x\, \to \,-\infty}⁡ \dfrac{ x^2+2x-3 }{ x^2+1 }=1$$ While making evaluations on my draft, i get: $|f(x)-L|= \dfrac{2}{|x-1|}$ I want to "remove" the absolut value in order to find ...
-1
votes
1answer
3k views

mean and median of an arithmetic progression

Is the mean always equal to the median of an arithmetic progression e.g. for a set of consecutive integers $x, x+1, x+2, \dots,y-2, y-1, y$ The median is $(x+y)/2$ equals the mean $(x + x+1 +\dots+ ...
-1
votes
1answer
57 views

Help understanding about Series and Sequences

I was hoping to see if anyone could help me out about explaining about Sequences and Series? Because I am getting bit stuck on how to really understand the concept of certain Sequences Here's an ...
-1
votes
2answers
2k views

Does the series $\sum_{n=0}^\infty\frac{\sin(n+\frac 12)\pi}{1+\sqrt{n}}$ converge?

Does the series $$\sum_{n=0}^\infty\frac{\sin(n+\frac 12)\pi}{1+\sqrt{n}}$$ This is supposed to be an alternating series but I can't seem to figure out what the $b_n$ is in this case. is there some ...
-1
votes
1answer
369 views

Arithmetic Sequence Word Problem

. Lucy Ango’yuaq, from Baker Lake, Nunavut, is a prominent wall hanging artist. This wall hanging is called Geese and Ulus. It is 22 inches wide and 27 inches long and was completed in 27 days. ...
-1
votes
1answer
30 views

Evaluate: $x_i\ge0; \max_{1\le i \le n , \sum_{i=1}^{n}{x_i=a}}x_1x_2…x_n$

Let a be a fixed positive real number.Evaluate: $$x_i\ge0; \max_{1\le i \le n , \sum_{i=1}^{n}{x_i=a}}x_1x_2...x_n$$ I guess it should be $x^n$ where $x=\frac{a}{n}$since this is true for $n=2$, I ...
-1
votes
1answer
34 views

counting gene sequences

A friend of mine was writing a paper arguing against allowing patents for genetic sequences. In one case, a company patented a 15-gene sequence. He asked mt how many 100-gene genomes contain this ...
-1
votes
1answer
142 views

Probability of ordered sequence

There are 3 squares, 5 triangles, and 4 circles. I need to generate possibilities of certain sequences if they are randomly generated. What is the probability that all the squares are grouped, next ...
-1
votes
1answer
62 views

Regarding the series representation of $\sin x \cdot\ln(1+x)$

In our calculus textbook, Thomas' Calculus, in Chapter 10.8, we are asked to find the first three nonzero terms of the Taylor approximation of the function $f(x) = \sin(x)\cdot\ln(1+x)$ as well as ...
-1
votes
1answer
49 views

McLaurin expansion

I've got an example and I wanted to know how it is expanded. Thanks for help. $$h''(x)+2h'(x)+h(x)=0$$ $$h(x)=5xe^{-x}+2e^{-x}$$ Is converted to $$h(x)={\sum _{n=0}^{\infty } \frac{5n{(-1)}^{n+1}x^...
-1
votes
1answer
74 views

How do I come to a series expansion of $1/(e^z-1)^2?$

How do I come to a series expansion of $\frac{1}{(e^z-1)^2}?$ $e^z-1$ can be expanded to: $$1 + z + \frac{1}{2} z^2 + \frac{1}{6} z^3 + \dots -1$$ so the series becomes: $$\frac{1}{(z^2 (1 + \frac{...
-2
votes
0answers
15 views

Given that $\sum_{R(j)} a_j$ is an infinite series, according to Knuth it can be represented as below

$$\sum_{R(j)} a_j = \biggl(\lim_{n\to\infty} \sum_{\substack{R(j)\\-n \le j \lt 0}} a_j\biggr) + \biggl(\lim_{n\to\infty} \sum_{\substack{R(j)\\0 \le j \lt n}} a_j\biggr)$$ I am reading The Art Of ...
-2
votes
1answer
52 views

Limit as n approaches infinity

Limit as n approaches infinity of $$ \left[\left(\frac{2\cdot 1+1}{2\cdot 1+3}\right)^{1^2}\cdot\left(\frac{2\cdot 2+1}{2\cdot 2+3}\right)^{2^2}\cdot \ldots \cdot\left(\frac{2\cdot n+1}{2\cdot n+3}\...
-2
votes
1answer
81 views

Is there a better way to calculate the Arc-Cosine-Hyperbolic

I am using $\operatorname{arcosh} x = \operatorname{arcsinh}(\sqrt { x^2-1 } ) $ with bad results. The standard power series $$ \operatorname{arcosh} x = \ln(2x) - \left( \left( \frac {1} {2} \...
-2
votes
2answers
22 views

sequence and series, limits

Struck with this expression while solving, can you help me solving this
-2
votes
1answer
32 views

sequence or not?

n= 1,2,3,4,5,6,...n c= 1,3,6,10,,15,21,...n(n+1)/2 m= 0,1,2,4,6,9... Am trying to find out if there a formulae that could be generated to fill in the sequence for m. I know the pattern as follows: ...
-2
votes
1answer
94 views

proof$\lim_\limits{n\to\infty}\left(\frac{\pi x_n -2}{ x_n}\right)=\pi -2$

$x_n$ is a sequence of real numbers that converges to 1 as $n\to\infty$ How to prove$$\lim_\limits{n\to\infty}\left(\frac{\pi x_n -2}{ x_n}\right)=\pi -2$$ by the formal definition of sequence ...
-2
votes
1answer
68 views

Is this series conditionally convergent or absolutely convergent? $\sum_{k=1}^{\infty}\left(-1\right)^{k+1}\sin\left(\frac{1}{k}\right)$

This series is not absolutely convergent because \begin{align*} \lim_{k\rightarrow+\infty}\frac{\bigl|\left(-1\right)^{k+1}\sin\left(\frac{1}{k}\right)\bigr|}{\frac{1}{k}} & =\lim_{k\rightarrow+\...
-2
votes
2answers
69 views

series 0, 6, 13, 20, 27

I'm working with days of the week and need to generate a series thats $n*7-1$ except for where $n=1$ where it should be $0$. So the series I'm after is $0,6,13,20,27$ I can't figure out a formula ...
-2
votes
1answer
69 views

Generatingfunctionology, why does nCk*y^n vanish when n is a negative integer?

In Generatingfunctionology (2nd ed available free here https://www.math.upenn.edu/~wilf/DownldGF.html ), nCk for integer n < 0 is defined as nCk = n*(n-1)...(n-k+1)/k! Then he asserts that: sum ...
-2
votes
1answer
422 views

Laurent series of $z^2\sin( \frac {1}{ z -1})$

I am having quite the trouble combining the coefficient of the Laurent series of $z^2sin(\frac{1}{1-z})$ at $0<\lvert z-1 \rvert<\infty$. At first, it seems pretty elementary since you can set $...
-2
votes
2answers
41 views

Infinite series convergence question

$$\sum_{n=3}^{\infty}\frac{(-1)^n}{\log n}$$ Can the conditional convergence of this series be proved by alternating series test, since you need n to be a natural number for the alternating series ...
-2
votes
1answer
45 views

convergence/divergence problem

Does the following series absolutely converge, conditionally converge or diverge? $$\sum_{n=1}^{+\infty}\sin(n^2)\sin\left(\frac{1}{n^2}\right)$$ I don't even know where to begin, I tried the limit ...
-2
votes
1answer
121 views

Question on the proof about simple harmonic series..

This is humble proof about harmonic series on my own. 1 + 1/2 + 1/3 + 1/4 + 1/5....... = 1 + (1 - 1/2) + {(1 - 1/2) - (1/2 - 1/3)} + {(1 - 1/2) - (1/2 - 1/3) - (1/3 - 1/4)} +... = 1 + (1/2)n - (1/6)...
-2
votes
1answer
34 views

Prove that $\frac{1}{\sqrt{1-\sin^2{x}}}=\sum\limits_{n=0}^{\infty} \frac{(2n)!(\sqrt{\sin{x}})^{4n}}{4^n (n!)^2}$.

Prove that $$\frac{1}{\sqrt{1-\sin^2{x}}}=\sum_{n=0}^{\infty} \frac{(2n)!(\sqrt{\sin{x}})^{4n}}{4^n (n!)^2}$$
-2
votes
1answer
46 views

Showing the limit exist by using sequences

$$a_1=1,a_{n+1}=\sqrt{6+a_n}, n \in \mathbb{N}$$ Show that the limit $\lim\limits_{n \to +\infty} a_n$ exists and find it. I know to prove that the sequence is bounded and monotonous but I still ...
-2
votes
2answers
62 views

Set closed. Hahn Banach theorem. Banach limits

I have a question: why is this set $A$ closed? $$A=\{x-x' \mid x \in l_\infty \}$$ Where $l_\infty$ is the set of bounded sequences, $$x=( x (1),x (2), x (3),\dots),$$ and $$x'=(x (2),x (3),x (4),\...
-2
votes
1answer
111 views

On calculations concerning the exact formula for the Mertens function, and the identity that relates Möbius and Mertens functions

I've deduced this identity involving the Riemann Zeta function $\zeta(s)$ and the Möbius function $\mu(n)$, the zeros and the meaning of how is understood the summation $\sum_{\rho}$ on assumption of ...
-2
votes
2answers
116 views

Self-similarity in mathematical sequences

Can any mathematical sequence be considered self-similar after some predefined number of terms similar to the initial sequence? Such a function could be something crazy or simply every second term, ...
-2
votes
1answer
71 views

Investigate the convergence of the series

Investigate the convergence of the series on the interval $x \in [0;1]$ $$\sum_{n = 1}^{\infty} \frac{x}{1 + n^2x^2} $$
-2
votes
3answers
162 views

Prove that the sum of 2 increasing sequences is also increasing.

Attempt 2: Let’s refer to a sequence $\{x_n\}_{n=1}^\infty$ as increasing after a while if the following is true: $(\exists N \in \mathbb N)(\forall n \in\mathbb N)$$(n \leq N$ $\rightarrow$ $x_{n+1}...
-2
votes
1answer
40 views

Can this sequence be expressed with a closed-form formula?

1, 3, 5, 7, 9, 13, 17, 21, 25, 31, 37, 43, 49… So if you don't count the first number the sequence is +2 for four times. After that it changes to +4 for four times, then +6, +8 and so on, all for ...
-2
votes
1answer
300 views

How can I find out if a non-convergent series is “indeterminate” (that is, “oscillating”) or “divergent”?

Definitions: Given a sequence $\{a_n\}$, define $$s_n= \sum_{j=0}^n a_j.$$ The sequence $\{s_n\}$ is called the series of partial sums of $\{a_n\}$. A series is convergent if $\{s_n\}$ has ...
-2
votes
4answers
60 views

Finding sum of two terms in a geometric progression

If $36, p, \dfrac 94, \mathrm{and}\,q$ are consecutive terms of a sequence, what is the sum of $p$ and $q$. I don't even know where to start.
-3
votes
2answers
51 views

What is the easiest approch to solve this sequences and series problem?

I've attempted this problem by counting pages, which is a tedious approach, is there a shorter method?
-3
votes
1answer
62 views

Use Taylor Theorem to find the polynomial approximation

A) Find a polynomial approximation for $f(x)=2e^x$ centered at $0$ for values of $x$ in the interval $[-1,1]$ B) what is the actual bound on the error in your approximation given by Taylor theorem? ...
-3
votes
1answer
39 views

Can you prove the following Cauchy sequence

Let (An) be a Cauchy sequence such that n is a member of N, and let c be a member of R, prove that (c*An) is also a Cauchy sequence
-3
votes
1answer
61 views

Finite Sum with Powers of Cosines

Please I have tried to show that 1) $\sum_{j=1}^{N-1}\cos^2(\frac{2\pi}{N}\cdot j\cdot\Delta c)=N-1 $ for $\Delta c=\frac{N}{2} $ and that 2)$\sum_{j=1}^{N-1}\cos^2(\frac{2\pi}{N}\cdot j\cdot\...