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
3
votes
1 answer

Calculate $\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(2^m n+1)}$

As title, to calculate $$\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(2^m n+1)}$$ I tried to calculate $$\sum_{m=0}^{\infty}\frac{1}{2^m n+1}$$ but to no avail. Then, as the addend looks similar to $$\ln(1+x) = \sum_{n=1}^{\infty}…
athos
  • 5,177
3
votes
1 answer

Where monotony is required of $\sum_{n=1}^{\infty}n(a_n-a_{n+1})$

Good day, solving this problem did not cause any problems, however, I don’t understand where the monotonicity of the sequence is required If the terms of a sequence $a_n$ are monotonic, and if $\sum_1^{\infty}a_n$ converges, show…
didenko jack
  • 321
3
votes
2 answers

Conjecture about a property of convex (concave) functions

Trying to prove a proposition in my paper, which can potentially use a conjecture about convex (concave) functions. I think the following is intuitive but have no idea how to rigorously prove it. Conjecture For a convex (concave) function…
Chang
  • 87
  • 6
3
votes
0 answers

Are there infinitely many integers $k$ such that $\sum_{n=0}^\infty \frac{n^k}{3^n}$ is also an integer?

The first few numbers in the function, represented as (k, f(k)) are as follows: (0, 1/2) (1, 3/4) (2, 3/2) (3, 33/8) (4, 15) (5, 273/4). I found these using the identity $\sum_{n=0}^\infty \frac{(n+m-1)!}{m!(n-1)!a^n} = \frac{a^m}{(a-1)^{m+1}}$ and…
LogicCube
  • 67
3
votes
1 answer

Evaluating the sum $\sum_{j \in \mathbb{Z}} (t 2^{2j})^s e^{- t 2^{2j}}$

How can I go about showing that $$ \sup_{t > 0} \sum_{j \in \mathbb{Z}} (t 2^{2j})^s e^{- t 2^{2j}} < \infty, \quad s > 0 $$ It doesn't seem like we can move the $\sup$ anywhere besides outside as the terms would be blowing up near $j =…
newbie
  • 307
3
votes
1 answer

Is the series $ \sum_{n = 1}^{\infty} \frac{\alpha \log n}{1 + (n + \alpha)^2} $ uniformly bounded for $\alpha \geq 0$?

Is it true that the series $$ \sum_{n = 1}^{\infty} \frac{\alpha \log n}{1 + (n + \alpha)^2} $$ is uniformly bounded for every $\alpha \geq 0$, i.e., does there exists a constant $C$ such that the series is bounded by $C$ for every $\alpha \geq 0$?…
Epsilon-Delta
  • 137
3
votes
1 answer

$\sum a_n^3$ converges implies that $\sum \frac{a_n}{n}$ converges?

$\sum a_n^3$ converges implies that $\sum \frac{a_n}{n}$ converges? I guess it is wrong, since we do not assume $a_n\geq 0$. But how to find an example? Clearly, $a_n$ should be sign-alternating, but such kind of $(-1)^n/n^{1/3}$ does not help.
xldd
  • 3,407
3
votes
0 answers

Analytic function expansion with Riemann Zeta coefficients.

Anyone have any ideas of how to progress with this sum? $$ \sum_{j = 1}^{\infty} \dfrac{f^{(j)}(a)}{j!} \left(\zeta(-(j-1)e^{i \pi t}) - \dfrac{1}{(j-1)e^{i \pi t} + 1} \right).$$ $f(x) \in O(x)$ as $x \to \infty$, and it is analytic, $t \in…
BBadman
  • 197
  • 1
  • 7
3
votes
2 answers

Proving $\sum_{n \geq 1} \frac{a_n}{1+ a_n}$ diverges provided $\sum a_n$ diverges and $(a_n)$ is decreasing and nonegative

Could someone tell me why this might a wrong solution? The solution looked nothing like mine Let $t_n$ be the partial sums of $\sum_{n \geq 1} \frac{a_n}{1+ a_n}$ $$t_n = (1-\frac{1}{1+a_1}) + (1-\frac{1}{1+a_2}) + \dots + (1-\frac{1}{1+a_n}) \\…
Lemon
  • 12,664
3
votes
1 answer

how to prove that$\sum_{n=1}^{\infty}{ n^4\over n!}=15e$

How to prove that $$\sum_{n=1}^{\infty}{ n^4\over n!}=15e$$ I think this is a problem of exponential series.
DebojyotiMukh
  • 915
3
votes
1 answer

Sum of $n^2/(n^2-a^2)(n^2-b^2)$

I'm trying to figure out what the sum of this series could be $$ \sum_{n=1,n\neq a,b}^{+\infty} \frac{n^2}{(n^2-a^2)(n^2-b^2)}\quad\text{with}\quad a,b\in\text{N}^+$$ but I've got no idea whatsoever: I tried differentianting or integrating in one of…
Rob Tan
  • 195
3
votes
2 answers

Tail sum of a monotonically decreasing sequence

Let $\{ a_k \}_{k = 1}^{\infty}$ be a non-increasing sequence of non-negative reals such that $\displaystyle \sum_{k = 1}^{\infty} a_k = 1$. Define $\displaystyle T_k := \sum_{l = k+1}^{\infty} a_l$. I am trying to show that the ratio $\dfrac{T_k}{k…
sudeep5221
  • 2,736
3
votes
2 answers

Find an alternating series such that its limit does not exist

I'm trying to find a sequence $a_n$ with $a_n\xrightarrow{}{0}$ and $a_n>0$, but such that $\sum_{n=1}^{\infty}(-1)^n{a_n}$ does not admit limit, or its limit is oscillating. From Leibniz's rule if $a_n$ satisfies the previous hypothesis and…
Roberto Palermo
  • 392
3
votes
4 answers

Find the sum of the series $1^2-2^2+3^2-4^2+...-(2n)^2$

Find the sum of the series $$1^2-2^2+3^2-4^2+...-(2n)^2$$ I tried rewriting it as $$\sum_{r=1}^{2n}-1^{n+1}(r^2)$$ but it didn't help. Also, looked at re-arranging as $$1^2+3^2+5^2+7^2+...+(2n-1)^2$$ and $$-2^2-4-6^2-8^2-...-(2n)^2$$ Still couldn't…
Gez Bishop
  • 771
3
votes
1 answer

A fast way to compute the infinite series:$\sum_{n=0}^{\infty}\frac{n^3((2n)!)+(2n-1)(n!)}{(n!)((2n)!)}$

I'm aware that I have to use the expansion for $e^x$ for this problem, ie: $e^x =\sum_{n=0}^{\infty} \frac{x^n}{n!}$. However the problem is with the computation, and it's getting too long, I'm trying to find a method which wouldn't take as much…
q123LsaB
  • 327
  • 1
  • 12
1 2 3
…
99 100