# 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.

44,468 questions
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### Prove $\forall m\in\mathbb{N},m\neq1:\quad\sum_{n=1}^{m}\frac{1}{n^2}\leq\int_1^m\frac{\sqrt{x^6+4}}{x^3}\ dx$

So I have stumbled upon this question and was very intrigued on how to solve it. I have an intuitive solution, but I guess that's not enough. I would be glad if you could shed some light on how to ...
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### Finding floor of reciprocal sum

Evaluation of $$\bigg \lfloor \frac{1}{\sqrt[3]{1}}+\frac{1}{\sqrt[3]{2^2}}+\frac{1}{\sqrt[3]{3^2}}+\cdots +\frac{1}{\sqrt[3]{(1000)^2}}\bigg\rfloor$$ Where $\lfloor x\rfloor$ is the floor ...
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### Limit of infinite product: (3n+1)/(3n+2)

I am struggling to find the following limit: $$\lim_{n\to \infty} \frac{1\cdot4\cdot7\cdot...\cdot(3n+1)}{2\cdot5\cdot8\cdot...\cdot(3n+2)}$$ So far I tried to use squeeze theorem and logarithm to ...
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### Equality of 2 forms of summation - derangement problem

The number of ways of placing $n$ objects not in position is given by the inclusion-exclusion number $D_n$: $n! \left( 1-\dfrac{1}{1!}+\dfrac{1}{2!}+....+(-1)^n\dfrac{1}{n!} \right)$ which can also ...
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### Digits of irrational numbers, organized chaos, modular arithmetic, and uniform equilibrium distribution

Is there a simple increasing sequence $\{ x_n \}$ of positive integers such that $x_n \mbox{ mod } b$ is the $n$-th digit of some known mathematical irrational constant in some integer base $b$? ...
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### Geometry / Calculus - relationship between surface area and volume of a burrito

Basically, trying to find the relationship between the radius of the surface and the volume of a burrito as a (graph-able?) mathematical progression in terms of radius (or width) of the tortilla. For ...
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### How to solve the recurrence relation $a_{1}=2, a_{n}=\frac{a_{n-1}+2}{2 a_{n-1}+1}(n \geq 2)$ with generating functions?

There's already a way to solve it, called "fixed point method", that is, from the relation we define its characteristic equation as $x=\dfrac{x+2}{2x+1}$，then we have $x_1=1,x_2=-1$. So the following ...
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### Prove that $\lim_{j \to\infty}\sum_{k = 1}^{\infty} \frac{a_k}{j+k} = 0$

Exercise: Suppose that $a_k \geq 0$ for $k$ large and that $\sum_{k = 1}^{\infty} \frac{a_k}{k}$ converges. Prove that $$\lim_{j \to \infty}\sum_{k = 1}^{\infty} \frac{a_k}{j+k} = 0$$ Attempt ...
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### How can I prove this sequence converges to 1?

Suppose $0<a<1$ and define $a_n=(1+a^n)^n$, show that $a_n \to 1$ by using binomial expansion on each $a_n$ and compare to a geometric sum. I know how to compute binomial on $a_n$ but I am ...
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### Combinatorics proof involving finite series

I am trying to prove the following identity with little success!: $\sum_{k=0}^{p-1} (p-k) = p(p+1)/2$. Any suggestions? Thanks.
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### What strategies can I adopt for proving monotonicity of a sequence. (Do not have a problem in mind but I am open to discussions)

So I have been thinking about different ways/strategies of proving monotonicity of sequences, as I said in the title, I do not have a problem in mind but I want to hear different ways I may use during ...
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### How to generalize the Thue-Morse sequence to more than two symbols?

The Thue-Morse sequence is defined as a binary sequence and can be generated like 0, 01, 01 10, 01 10 10 01, 01 10 10 01 10 01 01 10, ... . So the second ...
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### $Z = \sum_{1 \le r < s \le 2n}^{2n} f(r,s) x_{r} x_{s}$, How to prove the the maximum occurs at the end points?

We have the summation $$Z = \sum_{1 \le r < s \le 2n}^{2n} f(r,s) x_{r} x_{s}$$ The possibilities of $x_{i}$s are bounded by $-1 \le x_{i} \le 1$. Also, $f(r,s)$ real function. How to prove the ...
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### Is this Parentheses-Inserting, Series-Rearranging and Re-indexing argument correct?

Let $X$ be an infinite subset of $\mathbb{R}$, and let $\left\{ f_{n}\left(x\right)\right\} _{n\geq0}$ be a sequence of functions $f_{n}:X\rightarrow\mathbb{C}$ so that, for each $x\in X$, there is a ...
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### Is this series for Pi correct? And who has done it before?

The idea was to use an infinite series of triangles. The red then green then the... to get the area of this sector then the area of the circle is 16 times this. If it is a unit circle than area should ...
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### An investigation of Heine´s characterization of continuity

This characterization of continuity, for example, for functions $f: \mathbb R \to \mathbb R$, can be stated as: A function $f: \mathbb R \to \mathbb R$ is continuous at the point $x_0$ if and only ...
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### Two proofs concerning sequence [closed]

Let $S$ be a non-empty subset of $\mathbb{R}$ having a limit point $l$. Show that there exists a sequence $\left\{x_n\right\}$ of distinct elements of $S$ such that $\lim x_n= l$. Let S be an ...
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### How to find the growth rates of $n$ bacteria, knowing the sizes of bacteria from $m$ observations?

Each bacterium grows at a some constant rate, i.e. every minute the size of the bacteria increases by some constant value. Different bacteria can grow at different rate (they can also grow at same ...
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### Sum with floor of harmonic series [closed]

How can I prove the following? $$\sum_{i=1}^{n} \left\lfloor \frac{n}{i} \right\rfloor \sim n \sum_{i=1}^{n}\frac{1}{i}$$
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### Evaluate $\frac {1+\frac {2^2}{2!} +\frac {2^4}{3!}+\frac {2^6}{4!} +\dots}{1+\frac {1}{2!}+\frac {2}{3!}+\frac {2^2}{4!}+\dots}$

Evaluate the given series $$\dfrac {1+\dfrac {2^2}{2!} +\dfrac {2^4}{3!}+\dfrac {2^6}{4!} +....}{1+\dfrac {1}{2!}+\dfrac {2}{3!}+\dfrac {2^2}{4!}+....}$$ If we factor out $\dfrac {1}{2^2}$ from the ...
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### Convergence in p-adics and reals

I have been trying to solve three closely related problems about convergence of sequences in the p-adic numbers. I managed to solve two, but got stuck on the last one. Apart from trying to find a ...
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### How to show that $\sum_{k=0}^{\infty}{(k+1)z^k}=\frac{1}{(1-z)^2}$ for $z\in \mathbb{C}:|z|<1$

How to show that $\sum_{k=0}^{\infty}{(k+1)z^k}=\frac{1}{(1-z)^2}$ for $z\in \mathbb{C}:|z|<1$ I believe that I have to use the cauchy product? But how do transform the expression to be a product ...
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### What is the sum of the series? $\sum_{i=0}^{K+N-3}\left\lfloor \frac{i}{N-1} \right\rfloor$

What is the sum of the series given by $$\sum_{i=0}^{K+N-3}\left\lfloor \frac{i}{N-1} \right\rfloor$$ Where $N \in [2,100000)$ and $k \in [1,100000)$ Please help me find the first term and last term ...