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.

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Finding $a_n$ with $a_n = o(n \cdot \log(n))$ and not $O(n)$

Can you give me an example of a sequence $a_n$ ($n \in \mathbb N$) that satisfies the above conditions? $o$ and $O$ are Landau symbols.
collimarco
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Prove that $\arctan\Big(\frac{100 \log^2 n}{\sqrt{n+1} - \sqrt[3]{n}}\Big)$ is decreasing

I would like to prove that $\exists n_0$ such that the sequence $a_n = \arctan\Big(\frac{100 \log^2 n}{\sqrt{n+1} - \sqrt[3]{n}}\Big)$ is decreasing $\forall n \ge n_0$. It is sufficient to show that the sequence $b_n = \frac{\log^2 n}{\sqrt{n+1} -…
David
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Inequality on sequence

A sequence of $a_n, b_n$ is defined as follows. $a_1, b_1 >0$, and $$a_{n+1}=a_n+\frac{1}{b_n}$$ $$b_{n+1}=b_n+\frac{1}{a_n}$$ Prove that $a_{50}+b_{50}>20$ $$\begin{array}{l} \frac{a_{n+1}+b_{n+1}}{2}…
Ellie_Wong
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How to get an expression for Sum of first n terms of the following series?

$1^2+2\cdot2^2+3^2+2\cdot4^2+5^2+2\cdot6^2+\ldots$ I have tried it a number of ways First way is I split the series into halves like this $ ( 1 ^ 2 + 2 ^ 2 + 3 ^ 2 + 4 ^ 2 + 5 ^ 2 + 6 ^ 2 +...) + (2 ^ 2 + 4 ^ 2 + 6 ^ 2 +... ) $ Took last term of…
Manu Sm
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A summation involving the absolute value of $\sin x$

What would the sum $$\sum_{n=1}^{\infty} \frac{|\sin (n x)|}{n^2}$$ evaluate to? Without the modulus, Wolfram says that $$\sum_{n=1}^{\infty} \frac{\sin (n x)}{n^2}=\frac{1}{2} i\left(\mathrm{Li}_2\left(e^{-i x}\right)-\mathrm{Li}_2\left(e^{i…
Anomaly
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Check convergence of given series

$a_n≥0$ for all $n$ such that $a_n \to {\infty}$ as $n \to {\infty}$. Then there exist a natural number $M$ such that $\sum_{n=0}^{\infty}$$\frac{1}{(a_n) ^M}$ is convergent. (T/F) My solution: There exist $K$ such that $a_n>1$ for all $n≥K$. Choose…
Nope
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"Measuring" subsets of the natural numbers according to the convergence/divergence of the sum of their reciprocals

Let $S \subset \mathbb N$ and $f(S) = \sum_{x \in S} \frac{1}{x}$. Is there some notion of the 'size' of $S$ that lets us determine whether $f$ converges or diverges? Clearly, if $S$ is finite, $f$ converges. If $S = \mathbb N$, it diverges. But for…
The Infinite One
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Convergence of series using ratio test.

For the series $$\sum_{n=1}^{\infty}{\frac{n^2}{(n+1)(n^2+2)}},$$ application of the ratio test gave me a result of the limit equal to $1.$ This says that the test is inconclusive and should test for convergence using other ways. So I used the limit…
thuang
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How to determine if $\frac{2n-1}{3n+1}$ is bounded

I have to determine whether the above sequence is bounded (from above or below). Bounded from above means: $\frac{2n-1}{3n+1} \le M$ $\forall n$ Bounded from below means: $\frac{2n-1}{3n+1} \ge m$ $\forall n$ Then I tried to solve the above…
collimarco
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Prove all numbers in a sequence $a_{n+3}=15a_{n+2}-15a_{n+1}+a_n$ are perfect squares

$a_{n+3}=15a_{n+2}-15a_{n+1}+a_n$, here $a_1=a_2=1,a_3=9$ Prove all numbers in a sequence are perfect squares. My attempt is first to use the general formula of $a_n$. It is $a_n=\frac{1}{6}\left((2+\sqrt3)^{2n-3}+(2-\sqrt3)^{2n-3}+2\right)$ But I…
noname1014
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How to check whether this sum converge or diverge?

Consider the following sum: $$\sum_{k=1}^\infty\prod_{j=1}^k\frac{1}{\sqrt{j+1}-\sqrt{j}+1}$$ How could I check whether this sum converge or diverge? Root and ratio tests are inconclusive...
JohnWO
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For a Fibonacci-like sequence, $F_mF_n\ne F_p$

Let the sequence $\{F_n\}$ satisfy $F_1=a$, $F_2=b$ where $a,b\in\Bbb N_+$, and that $$F_{a+2}=F_{a+1}+F_a~~\forall a\ge1.$$ If $F_x$ is coprime with $F_y$, show that there doesn't exist a term $F_z=F_xF_y$. The simplest case is $a=b=1$. In this…
user1034536
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Proving that $\frac{2n}{n-2}$ is not an integer for all $n > 6$

I'm working on problem 1-6 in Exercises for the Feynman Lectures and the question asks: Can you explain why there are no crystals that have the shape of a regular pentagon? (Triangles, squares, and hexagons are common in crystal forms). This is what…
User13114
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Proving irrationality of $\sum_{n = 1}^\infty 2^{−2^n}$ with basic concepts

I've seen many posts about $$\sum_{n = 1}^\infty 2^{−2^n}$$ and many of them deal with irrationality of this limit. However, they all seem to try to prove its irrationality using theorems and irrationality tests like Liouville's theorem. I'm…
haha
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Convergence of a series $\sum_{n=3}^{\infty}\frac{1}{(\log\log n)^{\log n}}$

I wanted to test the convergence of the series $$\sum_{n=3}^{\infty}\frac{1}{(\log\log n)^{\log n}}.$$ First I, apply the Cauchy condensation test i.e., $\displaystyle\sum_{n=3}^{\infty}\frac{2^n}{(\log\log 2^n)^{\log2^n}}$ which is same as…
Bhukya
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