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 sum of a series: difference of cubes

I am trying to find sum of the infinite series: $$1^{3}-2^{3}+3^{3}-4^{3}+5^{3}-6^{3} + \ldots$$ I tried to solve it by subtracting sum of even cubes from odd, but that solves only half of the numbers. Any input is appreciated. Thank you all…
germyzz
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If $\frac {a_{n+1}}{a_n} \ge 1 -\frac {1}{n} -\frac {1}{n^2}$ then $\sum\limits_na_n$ diverges

Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that, for every $n\ge1$, $$\frac {a_{n+1}}{a_n} \ge 1 -\frac {1}{n} -\frac {1}{n^2} \tag 2$$ Prove that $x_n=a_1 + a_2 + .. + a_n$ diverges. It is clear that $x_n$ is increasing,…
user261263
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How many sequences of consecutive integers are there where the sum equals the length

I am really sad and I noticed that the sequence: $0 , 1 , 2$ Has its sum equal to its length. I was wondering how many these existed. e.g: $ 1$ $-3 , -2 , -1 , 0 , 1 , 2 , 3 , 4 , 5$ $(= 9)$ I got so far and got stuck, I reduced it down to finding…
Robert
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A positive sequence must monotonically converges for large enough index.

This question is rather simple, Let $a_n$ be a sequence that converges to zero, exists a $N$ such that for all $n>N$ the following $a_{n+1}\le a_{n}$ Is the theorem above correct? I am confused since I used it in an exam and the professor said that…
Rab
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How did Euler prove this identity?

While studying Fourier analysis last semester, I saw an interesting identity: $$\sum_{n=1}^{\infty}\frac{1}{n^2-\alpha^2}=\frac{1}{2\alpha^2}-\frac{\pi}{2\alpha\tan\pi\alpha}$$ whenever $\alpha \in \mathbb{C}\setminus \mathbb{Z}$, which I learned…
HyJu
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Prove that the elements of the sequence are integers

A sequence is given by the following recursion. Prove that the elements of this sequence are integers! $a_0=1$ $a_1=41$ $a_{n+2}=3a_n+\sqrt{8(a_n^2+a_{n+1}^2)}$
Sz_Z
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Show that $\sum\limits_{n=1}^\infty \frac {\sqrt{a_n}}{n}$ converges if $\sum\limits_{n=1}^\infty{a_n}$ does provided that $a_n>0$

Consider a convergent sequence $\sum\limits_{n=1}^\infty{a_n}$, with $a_n\ge 0$. Show that the series $$\sum\limits_{n=1}^{\infty} \frac {\sqrt{{a_n}}}{n}$$ also converges. Hint: $pq\le \frac 12(p^2+q^2)$
Applied mathematician
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Is it possible to use regularization methods on the Harmonic Series?

I recently learned about summation methods when dealing with divergent series to give them a finite value. An example of this isusing Cesàro summation on Grandi's series to get 1/2. However every method I know of is unable to sum the harmonic…
user6519
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Showing that $\sum_{n=0}^{\infty}\frac{2^{n+2}}{{2n\choose n}}\cdot\frac{n-1}{n+1}=(\pi-2)(\pi-4)$

Showing that (1) $$\sum_{n=0}^{\infty}\frac{2^{n+2}}{{2n\choose n}}\cdot\frac{n-1}{n+1}=(\pi-2)(\pi-4)$$ see here (2) $$(\arcsin(x))^2=\frac{1}{2}\sum_{n=1}^{\infty}\frac{(2x)^{2n}}{n^2{2n\choose n}}$$ Look very much similar to (1). How can I make…
user339807
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If the sum to 4 terms of a geometric progression is 15 and the sum to infinity is 16 find the possible values of the common ratio.

I can't find a way to get an answer for this. I have tried using the formula for the sum to infinity and dividing it by the sum to 4 terms but i can't get it to work.
cucucucu
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contradictory wikipedia and mathworld.wolfram

On https://en.wikipedia.org/wiki/Liouville_function there is written $L(n)>0.06 \sqrt n$ and $L(n)<-1.39 \sqrt(n)$ for infinitely many $n$. On http://mathworld.wolfram.com/LiouvilleFunction.html they say it is unknown if $L(n)$ changes sign…
Stijn Cambie
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Series of squares of n integers - where is the mistake?

Given the following two series: $$1^3 + 2^3 + ... + n^3$$ $$0^3 + 1^3 + .... + (n-1)^3$$ I take the difference vertically of the two: $$\left(1^3-0^3\right) + \left(2^3-1^3\right) + .... + \left(n^3-(n-1)^3\right)$$ This equals to $n^3$ If I now…
Naz
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$ S_{n}=\frac{x}{x+1}+\frac{x^2}{(x+1)(x^2+1)}+...........+\frac{x^{2^{n}}}{(x+1)(x^2+1)...(x^{2^{n}}+1)}$

If $\displaystyle S_{n}=\frac{x}{x+1}+\frac{x^2}{(x+1)(x^2+1)}+\frac{x^{2^{2}}}{(x+1)(x^2+1)(x^{2^2}+1)}+...........+\frac{x^{2^{n}}}{(x+1)(x^2+1)...(x^{2^{n}}+1)}$ Then $\displaystyle \lim_{n\rightarrow \infty}S_{n} = \;,$ Where $x>1$ $\bf{My\;…
juantheron
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Calculate $\ln(2)$ using Riemann sum.

Possible Duplicate: Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$? Show that $$\ln(2) = \lim_{n\rightarrow\infty}\left( \frac{1}{n + 1} + \frac{1}{n + 2} + ... + \frac{1}{2n}\right)$$ by considering the lower Riemann…
stariz77
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Value of $\sum_{n=0}^\infty (-1)^n$

I saw this equation on a blackboard today: $\displaystyle \sum_{n=0}^\infty (-1)^n$ It got me thinking -- this must oscillate between $1$ and $0$, yes? So then does this sum even have a meaningful value?
Peter
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