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 series $\sum_{n=1}^{\infty} a_n$ with $|a_n| \le \frac{n+1}{n^2}$

Let us consider series: $$S=\sum_{n=1}^{\infty} a_n$$ $$a_n=\frac{n\cos(n)+\sin(n^2+2n-5)}{n^2}$$ The bounds for $a_n$ is shown in this MSE question to be: $$|a_n| \le \frac{n+1}{n^2}$$ Is it possible to prove that this series is…
mike
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Non-zero terms in a series

Suppose I know that |$\sum_{k=0}^\infty a_kz^k$| converges to $0$ as $|z| \to \infty$, and $\sum_{k=0}^\infty a_kz^k$ converges absolutely for every $|z| >1$ ($z$ is complex). Is it true that all the $a_k$ are equal to zero. This seems to be so but…
jpv
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Telescoping property: $\sum_{k \geq 0}\frac{1}{(4k+1)(4k+3)}$

I need to calculate the sum $$\sum_{k \geq 0}\frac{1}{(4k+1)(4k+3)}$$ I've made some attempts to transform this in a summation that I could apply the telescopic property, but I didnt have any success. Thanks in advance!
Giiovanna
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Finding a formula for a given series.

I'm having trouble figuring out how to evaluate: $$\sum _{j = 1}^{n} j!j$$ I've tried plugging in numbers and looking for a pattern and I've also tried to find a general form like: $$n=1:$$ $$ 1*1 = 1 = n^2$$ $$n =2:$$ $$ 1*1 + 1*2*2 = 5 = (n-1)^2…
Pandamonium
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Evaluation of $\sum n a^n$ using telescoping property

Show that the series $$\sum\limits_{n=1}^{\infty}n a^n = \frac{a}{(a-1)^2} $$ for $|a|<1$ using the telescoping property. I know how to do this using other methods. But the exercise asks to use telescoping property.
Giiovanna
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Closed form of a complex series sum

I am working on a proof that require a closed form (if that is not possible then at least a tight lower bound) of the expression below: $$A(n,k)=\sum_{i=1}^k \left(1-d^{-i}\right)^{n-1}\left(\prod_{j=1,j \neq i}^k\left( 1- d^{j - i}\right)…
Indranil Banerjee
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How prove $T_{n}\neq 0$,if $T_{n+2}=(1-2c)T_{n+1}+(2c+a-c^2)T_{n}-(a-c^2)T_{n-1}$

Question: Assmue that the postive integer $a,c$ ,such $\lfloor \sqrt{a} \rfloor=c$ ,Now let sequence $$y_{1}=1,y_{2}=-2c, y_{n+2}=-2c\cdot y_{n+1}+(a-c^2)y_{n},n\ge 1$$ show that $$T_{n}=y_{1}+y_{2}+\cdots+y_{n}\neq 0,\forall n\in N^{+}$$ where…
math110
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Convergence of Newton series

What is the condition for a real valued function of a real variable to have a Newton series which converges to that function pointwise? It feels like there should be a condition similar to that for the Taylor series.
user157872
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How do I use Euler's result to find the sum of a series?

So I am given: $$ \zeta(4) = \sum_{n=1}^\infty {1\over n^4}={\pi^4 \over 90} $$ I need to use it to find the sum of the following series using the above information. $$ \sum_{k=1}^\infty {1\over{(k+2)^4}} $$ So, this is what I have so far: $$…
Oyukyfairy
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Upper bound of $\sum_{j=1}^p \frac{p+1}{p-j+1} \frac1{2^j}$

I am looking for an upper bound of the following sum $$ S_p:= \sum_{j=1}^p \frac{p+1}{p-j+1} \frac1{2^j}. $$ The upper bound should be independent of $p$, of course. Numerical experiments indicate that $$ S_p \le \frac53 $$ with the maximum…
daw
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Even-Odd pair in a sequence

Suppose we have a sequence of $n$ integers not necessarily distinct. Let's define, $E$ = Number of pairs $(i, j)$ such that $i
savvi singh
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Property of Subadditive Sequence

Call a sequence $\left \{ a_n \right \}$, $n \geq 1$, strictly subadditive if it satisfies the inequality $$ a_{n+m} < a_n+a_m $$ for all $m$ and $n$. I am wondering whether it is necessarily true that a positive strictly subadditive sequence…
user137147
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Are there any 3 natural numbers that satisfy $a^2+b^2=2z^2 $?

Are there any 3 natural numbers that satisfy $a^2+b^2=2z^2 $? This is a question that arised as I was trying to solve another question: Is there an arithmetic progression, of natural numbers in which three (not necessarily successors) elements…
user83081
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Divergence of a series (Raabe fails)

Consider $\displaystyle a_n=\left(\frac{(2n)!}{2^{2n}(n!)^2}\right)^2$ Prove that $\sum a_n$ diverges Lots of factorials, so first thing is to check for ratio test (fails), Raabe test (also fails). I can't find any lower bound that goes to…
Gabriel Romon
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Convergence of infinite series involving $\frac{\sin(x)}x$.

Show that the infinite series $$\sum_{x = 1}^{\infty} \frac{\sin(x)}{x}$$ is convergent. Please answer so that a Calculus student can understand.
Sally G
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