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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Is there a general formula for the sum of a quadratic sequence?

I tried Googling "formula for sum of quadratic sequence", which did not give me anything useful. I just want an explicit formula for figuring out a sum for a quadratic sequence. For example, how would you figure out the sum of $2+6+12+20+\dots+210$?…
Anonymous Computer
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Theorem 3.50 - Principles of Mathematical Analysis - Rudin

Theorem Suppose $$\begin{align}(a)&\sum_{n=0}^\infty a_n\quad\text{converges absolutely}\\(b)&\sum_{n=0}^\infty a_n=A\\(c)&\sum_{n=0}^\infty b_n=B\\(d)&c_n=\sum_{k=0}^n a_k b_{n-k}\quad(n=0,1,2,\dots)\end{align}$$ Then $$\sum_{n=0}^\infty…
Charlie
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Is there any function that has the series expansion $x+x^{\frac{1}{2}}+x^{\frac{1}{3}}+\cdots$?

$$\frac{1}{1-x} = 1+x+x^2+x^3+ \cdots$$ Is there a $f(x)$ that has the series of $n$th roots? $$f(x)= x+x^{\frac{1}{2}}+x^{\frac{1}{3}}+ \cdots$$ Wolfram Alpha seemed to not understand my input.
zerosofthezeta
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If $\sum p_n = 1$, can $\sum np_n$ be unbounded?

This question might seem quite easy and obvious, anyhow, I intend to make it clearer since this is just the beginning of a solution of a more general question. Given $\sum_{i=0}^\infty p_i=1$ and $p_i>0$ What is the answer to this…
Majid Biazaran
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computing trigonometical

I have problem with the divergence of a Sum, I don´t know how criteria use when i has a trigonometrical function because, it´s not monotone, and I can´t prove that this series diverge with the usual methods $$ \sum {\frac{{\cos \left( {\log \left(…
Daniel
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Convergence of double sum $\sum_{m, n}\frac{1}{m^p + n^k}$

For which ${k, p \in \mathbb{R}}$ does $\sum_{m,n \in \mathbb{N}} \frac{1}{m^p + n^k}$ converge? It is necessary but not sufficient that $p, k \gt 1$ I am looking for a simple solution, as the solutions I have seen so far have been rather perverted.
Mark
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Nice problem Which is bigger $e^a$ and $a^3$

let $$a=\sum_{n=0}^{\infty}\dfrac{\left(\dfrac{n+1}{3}\right)^n}{(n+1)!}$$ My Question: Which is bigger $e^a$ and $a^3$ I guess $$e^a=a^3$$ but I can't prove it,and I think this is nice problem.Thank you evryone This problem from this…
math110
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Some formulae for a periodic sequence $-1,-1,1,1...$?

Some formulae for a periodic sequence? when $T = 2$, we have $-1,1,-1,1,-1,1,\text{...}$, the formula is $$\begin{align*}(-1)^n\end{align*}$$ when $T = 4$, we have $-1,-1,1,1,-1,-1,1,1\text{...}$, the formula is? And how about the case $T=k$?
Sequence
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What should a 21st century Euler attempt?

Euler at the start of his career found the exact sum of the series $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$. My question is: What could a 21st century Euler possibly attempt to solve? Are there any similar "elementary" problems which a…
user92570
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What counts as a sequence, and how would we know that it isn't deceiving?

A sequence, in my understanding is just a list of numbers in order. In that case, if I write down random numbers with no pattern at all except for the fact that it gets larger, is it a viable sequence? In that case, given a graph, how can we tell if…
Hou Zeng
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Find the value of $1996^2-1995^2 + 1994^2-1993^2 + \dots + 2^2-1^2.$

Find the value of $$1996^2-1995^2 + 1994^2-1993^2 + \dots + 2^2-1^2.$$ What is wrong with the following reasoning. By the difference of squares formula we have that the sum can be broken into $$(1996-1995)(1996+1995) + (1994-1993)(1994+1993) +…
Louie
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proof for the Ramanujan's formula ?

I found this formula in a textbook in which the proof to the formula was not given Ramanujam's formula $$\sqrt{1 +n\sqrt{1 +(n+1)\sqrt{1 + (n+2)\sqrt{1 + (n+3)\sqrt{1 +....\infty}}}}} = n+1$$ Its a great equation andhow do you prove this. its a bit…
Suraj M S
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Better upper bound for $ \sum_{k=2}^{\infty} \frac{1}{2^{k-1}} \sum_{n=1}^{\infty} (n! \: \text{mod} \: k) $

Since the sum $\sum_{n=1}^{\infty} (n! \: \text{mod} \: k)$ will be zero beyond $k-1$, the series could be interpreted as an finite sum of length $k-1$. Also, the max value of $n! \: \text{mod} \: k$ is naturally $k-1$. Taken together one has a max…
DecarbonatedOdes
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An article from AMM vol 95 Page 942. (I have some doubt about it)

Today I am reading an article in American Math Monthly volume 95 Page 942. Author introduces an alternating series:$$\sum_{n=1}^{\infty}(-1)^n\frac{(2n)!}{4^n(n!)^2}$$ then he uses the Stirling's formula and alternating series test to conclude that…
Laura
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Can I use the distributive law to write $\sum_{i=0}^{\infty}2^i$ as $1+ 2\sum_{i=0}^{\infty}2^i$?

I've come across this problem: $$ S=\sum_{i=0}^{\infty}2^i \\ S= 1+ 2\sum_{i=0}^{\infty}2^i \\ S=1+2S \\ S=-1 $$ I know that you cannot rearrange this series because it is not absolutely convergent, but that seems to only happen from step 3 to 4 and…
Deedit
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