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
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Can a series be empty?

It sounds contra-intuitive, since it won't exist without elements. However, I am thinking in empty sets, null sequences and empty lists, which indeed exists.
xyz
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Series$\sum_{n=1}^\infty\big((1+1/n)^n-e\big)$

I can't figure out whether this series is convergent. I'm trying to use d'Alembert or Cauchy ratio tests, but however far I go with Taylor series it always ends up being one. The series is : $$\sum_{n=1}^\infty \big((1+1/n)^n-e\big)$$
James Well
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How can find this sequence $ a_{n+1}=a_{n}+na_{n-1},$

let $a_{n+1}=a_{n}+na_{n-1},a_{1}=1，a_{2}=2$. find the $a_{n}=?$ my ideas: $\dfrac{a_{n+1}}{(n+1)!}=\dfrac{1}{n+1}\dfrac{a_{n}}{n!}+\dfrac{1}{n+1}\dfrac{a_{n-1}}{(n-1)!},$ and let $b_{n}=\dfrac{a_{n}}{n!}$,then we have…
math110
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Sum of Infinite series with a Geometric series in multiply

I came across these series while solving a probability question. enter link description here let |r| < 1 , $$S_n=\sum_{k=0}^{\infty}k^n.r^k$$ For n=0 ,it's a GP. $S_0=\frac{1}{1-r}$ For n=1 ,it's a AGP , $S_1=\frac{-1}{1-r}+\frac{1}{(1-r)^2}$ for…
Rishi
  • 998
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Does $\sum_{k=1}^{\infty} \frac{1}{8^k+2^k+1}$ have a closed form?

Is there a closed form for this infinite summation? $$\sum_{k=1}^{\infty} \frac{1}{8^k+2^k+1}$$
John the Apostle
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Evaluate $\lim\limits_{n \to \infty}\left(\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots+\frac{1}{1+a_n}\right).$

Problem Let $a_1=3,a_{n+1}=a_n^2+a_n(n=1,2,\cdots)$. Evaluate $$\lim_{n \to \infty}\left(\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots+\frac{1}{1+a_n}\right).$$ Attempt Notice…
mengdie1982
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A riddle involving series.

Father has left to his children several identical gold coins. According to his will, the oldest child receives one coin and one-seventh of the remaining coins, the next child receives two coins and one-seventh of remaining coins, the third child…
Random
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Alternate complex binomial series sum

Calculation of $\displaystyle \sum^{2n-1}_{r=1}(-1)^{r-1}\cdot r\cdot \frac{1}{\binom{2n}{r}}$ is My Try: Using $$\int^{1}_{0}x^m(1-x)^ndx = \frac{1}{(m+n+1)}\cdot \frac{1}{\binom{m+n}{n}}$$ So $\displaystyle…
DXT
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Prove closed form for $\sum_{n\in\Bbb N}\frac1{5n(5n-1)}$

While looking for solutions to a difficult geometric problem, I encountered this sum: $$ \sum_{n\in\Bbb N}\frac1{5n(5n-1)} = \frac1{4\cdot 5} + \frac1{9\cdot 10} + \frac1{14\cdot 15} + \ldots $$ A bit of numerical exploring has convinced me that the…
Mark Fischler
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Is $\cot(\cot(\cot(\cdots\cot1)\cdots))$ always defined?

Consider sequence $a_1=1$, $a_{n+1}=\cot a_n$. Is $a_n$ always defined? Numerical evaluation suggests this conjecture is true. I have proved a weak version of this question: for a fixed $n$ and $a_1=x$, the measure of $x$ such that $a_n$ is not…
Kemono Chen
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Calculating the series $1/8+1/88+1/888+....$

I wonder whether this series is calculable or not. Attempt: $S=1/8+1/88+1/888+....=\dfrac18\displaystyle\sum_{k=0}^\infty\dfrac{1}{\sum_{n=0}^k10^n}$ where…
Micheal Brain Hurts
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Hard Telescoping Series

Finding the explicit sum of a telescoping series with two factors in the denominator is quite easy: we split the fractions in the difference of two subpieces. But what about 2+ factors? E.g., consider: $$\sum\frac{1}{(2n+1)(2n+3)(2n+5)}$$ We could…
MadHatter
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Prove that $ \frac{n^2}{\sum_1^n{\frac{1}{a_i}}} $ is convergent.

Suppose that $a_n > 0$ for all $n\geq 1$, and define $S_n = \sum_{i=1}^n{a_i}$. If $S_n$ is convergent, prove that $$ \frac{n^2}{\sum_{i=1}^n{\frac{1}{a_i}}} $$ is also convergent. Thanks.
Jichao
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Series with $\sum a_n$ converges but $\sum n a_n^2$ diverges, and $a_n$ is decreasing

Is it possible to have a sequence $a_n \geq 0$ which is decreasing and such that $\sum a_n < \infty$ but $\sum n a_n^2 = \infty$? I have seen Find a sequence $a_n$ so that $\sum |a_n|$ converges but $\sum n |a_n|^2$ diverges where the condition…
nullUser
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nth term of sequences

I'm a Phd student who teaches part time at a high school and I noticed something when teaching sequences today. I asked my students to find the nth term (the general term) for some sequences. They observed: If $a_n=n$ then the first differences…
Jack
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