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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Convergence of the series $\sum\limits_{n=3}^\infty (\log\log n)^{-\log\log n}$

I am trying to test the convergence of this series from exercise 8.15(j) in Mathematical Analysis by Apostol: $$\sum_{n=3}^\infty \frac{1}{(\log\log n)^{\log\log n}}$$ I tried every kind of test. I know it should be possible to use the comparison…
Charlie
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Find the sum, if exists $\sum\limits_{n=1}^{\infty} \frac{(2n)!}{2^{2n}(n!)^2(n+1)}$

$$ \sum\limits_{n=1}^{\infty}\dfrac{(2n)!}{2^{2n}(n!)^2(n+1)} $$ By comparison test this series converges. Any nice way to work the sum? I see that this can be written as: $$ \sum\limits_{n=1}^{\infty}\dfrac{\binom{2n}{n}}{2^{2n}(n+1)} $$
rrr
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Solve the "two trains and a fly" problem the hard way

Few days ago, I was asked the following question: There are $2$ cities. city $A$ and city $B$ with distance $d$=600km There are $2$ trains with speed of $vt$ = 100km/h. There is $1$ fly with speed of $vf$ = 300km/h. The question: Train 1 goes from…
d_e
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How find the value of the $a$ such $\{a^n\}$

Let us define the sequence $$a_{n}=\{a^n\}$$ with $a > 0$ where $\{a^n\}$ denotes the fractional part. How could we show that there is no positive real numbers $a\in Q$ such that this sequence $a_{n}$ is strictly increasing. I can find example…
math110
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How come $\sum\limits_{n=1}^{\infty}\frac{1}{2^n}=\sum\limits_{n=1}^{\infty}\frac{1}{2^n\ln(2^n)}$?

I'm a bit puzzled with the following: $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{2^n}=1$ $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{2^n\ln(2^n)}=1$ Which essentially yields the…
barak manos
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Sum of series with triangular numbers

Can you please tell me the sum of the seires $ \frac {1}{10} + \frac {3}{100} + \frac {6}{1000} + \frac {10}{10000} + \frac {15}{100000} + \cdots $ where the numerator is the series of triangular numbers? Is there a simple way to find the…
Mustafa Saad
  • 69
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Finding closed form for $1^3+3^3+5^3+...+n^3$

I'd like to find a closed form for $1^3+3^3+5^3+...+n^3$ where $n$ is an odd number. How would I go about doing this? I am aware that $1^3+2^3+3^3+4^3+...+n^3=\frac{n^2(n+1)^2}{4}$ but I'm not too sure how to proceed from here. My gut feeling is…
Trogdor
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A q-series identity coming out of Ramanujan Integral

While trying to solve a Ramanujan integral, it seems that we need to establish this $q$-series identity: $$1 + \sum_{n = 1}^{\infty}\frac{(-1)^{n}q^{n^{2}}}{(1 - q^{2})(1 - q^{4})\cdots(1 - q^{2n})} = \frac{(1 - q^{2})(1 - q^{4})(1 - q^{6})\cdots}{1…
Paramanand Singh
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The sum of reciprocal squares: estimating the remainder

Let $a_n$ denote the $n$th remainder of the series $$ 1+\frac{1}{2^2}+\frac{1}{3^2}+\ldots $$ In other words, $$ a_n = \frac{\pi^2}{6}-\left(1+\frac{1}{2^2}+\ldots +\frac{1}{n^2}\right). $$ I noticed that for small $n$ the following is true…
Anvita
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Does the series $\sum_{n=1}^\infty \left|\frac{\cos2^n}{n}\right|$ converge or not?

$$\sum_{n=1}^\infty~\left|\frac{\cos2^n}{n}\right|$$ I just confused what to do. I am sorry, i don't understand. But this task is important for me. Can you give full solution?
Simankov
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Infinite Series with factorial

I'm having trouble manipulating the function of this series which has factorials to show that it converges or diverges using the ratio test. The series is $\sum\limits_{k=1}^{\infty}\dfrac{(k!)^2}{(2k)!}$. The following are the steps I used, but I…
Sabien
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Cesàro sum of $\sum\limits_{n = 0}^\infty {\cos n}=\dfrac{1}{2}$. Please check my work

Thanks to the formula http://functions.wolfram.com/ElementaryFunctions/Cos/23/01/0001/ Partial sums $$s_m=\sum\limits_{k = 0}^m {\cos k = } \frac{{\sin \left( {\frac{1}{2}\left( {m + 1} \right)} \right)\cos \frac{m}{2}}}{{\sin…
Raffaele
  • 26,371
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Find limit of this decreasing sequence

$$a_n=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right) \cdots \left(1-\frac{1}{n^2}\right) $$ I have proved that this sequence is decreasing. However I am trying to figure out how to find its limit.
GorillaApe
  • 1,071
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Sequence sum question: $\sum_{n=0}^{\infty}nk^n$

I am very confused about how to compute $$\sum_{n=0}^{\infty}nk^n.$$ Can anybody help me?
user16653
  • 61
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Sum of polynomial

If $p(x)$ is a polynomial of degree $m$, does the polynomial $q(x)$ of degree $m+1$ exist so that $\sum_{i=0}^{n}p(i)=q(n)$? And if so, is there an algorithm to find the expression for $q(x)$?
Jostein Trondal
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