Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

Sequences and series are often considered as a main part of part of calculus, in addition to limits, continuity, differentiation and integration.

A sequence is an enumerated collection of objects in which repetitions are allowed. There are special types of sequences, such as arithmetic sequences (or arithmetic progressions), where the next term is a constant more than the previous; harmonic progressions, which is formed by taking the reciprocal of each term of an arithmetic progression; logarithmic progression, which is formed from a series whose progression getting smaller; and geometric progressions, where the next term is a constant multiplied by the previous term.

A sequence can be given by a direct formula (e.g. $a_n = 2^n + 3$), or by a recurrence relation. In a recurrence relation, the relation between the next term and the earlier terms is given. An example is the recurrence relation $F_{n+2}=F_{n+1}+F_n, n \geq 0.$ Together with the initial terms $F_0=0$ and $F_1=1$, this recurrence relation defines the famous Fibonacci sequence.

A series is formed by summing a sequence. A typical question is: When the number of summed terms goes to infinity, does the sum approach a finite limit? In other words, is it convergent? Several tests, such as the ratio test, the root test, the limit comparison test, the integral test, etc. can help to answer these questions.

Important note. Questions about guessing the next number in a sequence, with no explicit mathematical context, will usually be quickly closed. Consider posting these questions at Puzzling Stack Exchange instead.

65378 questions
4
votes
1 answer

How to evaluate infinite series $\sum\limits_{n=0}^\infty\sqrt{B^2+n^2} e^{-an}$

I'm trying to evaluate an infinite series: $$ \sum\limits_{n=0}^\infty\sqrt{B^2+n^2} e^{-an} $$ where $a$ and $B$ are real parameters, or equivalently: $$\sum\limits_{n=0}^\infty\sqrt{B^2+n^2} x^n$$ where $0
Ansrew
  • 41
4
votes
2 answers

What's special about the Cauchy product?

I am reading chapter 3 in Rudin's Principles of Mathematical Analysis. In it, Rudin defines the Cauchy product of two series. That is, given $\sum a_n$ and $\sum b_n$, $c_n=\sum_{k=0}^n a_k b_{n-k}$ and $\sum c_n$ is defined as the product of the…
daniel
  • 303
4
votes
1 answer

Why is this sum manipulation wrong?

Let $(a_n)$ with $a_n \in \mathbb{R}$ be such that $$\sum_{k = 0}^{\infty}\sum_{r = 0}^{2k}\Bigl|\frac{(-1)^k(2\pi)^{2k} a_r}{(2k - r + 1)!}\Bigr|< \infty$$ Why is following wrong? (Or is it correct?) $$\sum_{k = 0}^{\infty}\sum_{r =…
Reactant
  • 677
4
votes
0 answers

When can you add two infinite series term by term?

Let $a= \sum_{n \geq 1}a_n $and $b=\sum_{n \geq 1}b_n$. When can I say that $$a+b = \sum_{n \geq 1}a_n + b_n$$ ? What if all the terms were positive? Or do I need some absolute convergence?
QuestionAnswer
  • 153
4
votes
4 answers

Expressing the infinite sum $1 + 22 + 333 + 4444 + \dotsb$

I would like to express $$1+22+333+4444+\cdots$$ using $\Sigma$ notation, and have no clue where to start. After $999999999$, comes 10 $0$s, then 11 $1$s.
AAron
  • 413
4
votes
1 answer

Rationality of subseries

Let $(a_n)_n$ be a real sequence. Assume than $\sum_{n=1}^\infty a_n$ converges to a rational number. Does there exist a subseries which converges to an irrational constant? Assume now the opposite situation, I.e. the full series converges…
Angelo Rendina
  • 1,893
4
votes
0 answers

Why does $\sum_{n = 0}^\infty \left[ \frac{1}{(2z + 2n)^2} + \frac{1}{(2z + 2n + 1)^2} - \frac{1}{(2z + n)^2} \right] = 0$?

I am reading a textbook, and it uses the equation $\sum_{n = 0}^\infty \frac{1}{(2z + 2n)^2} + \sum_{n = 0}^\infty \frac{1}{(2z + 2n + 1)^2} = \sum_{m = 0}^\infty \frac{1}{(2z + m)^2}$. Could someone tell me why this is true? I am pretty…
pseudoname123456
  • 101
4
votes
1 answer

Series and alternating series of the same function that converge to inverse quantities

I know that from the definition for exponential we have that $e^x=\sum_{k\ge 0}\frac{x^k}{k!}$. As a consequence $$e=\sum_{k\ge0}\frac{1}{k!}\quad\quad \text{and}\quad\quad \frac{1}{e}=\sum_{k\ge0}\frac{(-1)^k}{k!}$$ So Im curious about if exist…
Masacroso
  • 30,417
4
votes
1 answer

Does $\displaystyle \sum_{n>0} \frac{1}{\sin n} $ converge?

Given the series: $$\sum_{n=1}^\infty \frac{1}{\sin n};$$ does it converge? I think not but I believe that I saw somewhere in a book that it does? .... or maybe that was $\displaystyle\sum\limits_{n=1}^\infty \frac{1}{{\sin n}^2}$? Edit: sorry, I…
Eiyrioü von Kauyf
  • 275
4
votes
3 answers

Uniform convergence: $\sum \frac{x^2}{(1 + x^2)^n}$

Consider the series of functions: $$\sum_{n \ge 1} \frac{x^2}{(1 + x^2)^n}$$ Q: Where does this series converge uniformly? We have if $x \neq 0$: $$\lim_{n \to \infty} \left| \frac{x^2}{(1 + x^2)^{n+1}} / \frac{x^2}{(1 + x^2)^n} \right|=…
user230734
4
votes
0 answers

Upper bounding Lerch zeta function

Let $\Phi\left(z,s,a\right) $ be a Lerch Trascendent. $$\Phi\left(z,s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{t^{s-1}e^{-at}}{1-ze^{-t}}dt.$$ Can we upper bound the above in terms of its parameters (for positive reals $z, a$…
Ram
  • 277
4
votes
1 answer

Convergence of $ \sum\limits_{n=2}^{\infty} \dfrac{1}{n^p \ln(n)^q} $

Let $p,q \in \mathbb{R}$. Show using the comparison test (or limit comparison test) that $$ \sum\limits_{n=2}^{\infty} \dfrac{1}{n^p \ln(n)^q} $$ converges for $p>1$ and any value of $q$ and that it diverges for $p<1$ and any value of $q$. My…
Lundborg
  • 1,646
4
votes
1 answer

Convergence of $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{\sin(n-m)}{n^2+m^2}$

Is the following series convergent or divergent? $$\displaystyle{ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{\sin(n-m)}{n^2+m^2}}$$ Even if it converge I do not know to prove it. However, for example, I know that $$\sum_n \frac{\sin…
Mark
  • 7,841
  • 6
  • 38
  • 72
4
votes
2 answers

Proving that $x^n$ converges to $0$ whenever $|x| < 1$

I've already proved that for any $p > 0$ and for any $\alpha \in \mathbb{R}$, the sequence $\frac{n^\alpha}{(1 + p)^n}$ converges to $0$. Now, I want to prove that $\lim_{n \to \infty} x^n = 0$ as long as $|x| < 1$. I've split it into two possible…
jamaicanworm
  • 4,494
4
votes
2 answers

Proving the convergence of $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$ for $|z| > 1$

$\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$, $|z|>1$. There are two facts that my professor uses that I am confused about. The first is: $|1+z^n| \geq ||z|^n-1|$, I believe this is true for any $|z|$. The other is: $\frac{1}{|z|^n-1} \leq…
Johnver
  • 1,961
1 2 3
…
99 100