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.
49
questions
-4
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Finding the limit for a sequence
Can you calculate the exact value of the limit of the numerical sequence (U) defined by U(0)=1 and
U(n+1) = ((e^n) / (1+e^n)).U(n)?
1
vote
2
answers
77
views
The relationship between $\sum_{n=1}^{\infty}\frac{b_n}{a_n}$ and $\sum_{n=1}^{\infty}\frac{b_1+\cdots +b_n}{a_1+\cdots+a_n}$
Let $a_n$ and $b_n$ be non-constant positive sequences. Given that the series $\sum_{n=1}^{\infty}\frac{b_n}{a_n}$ converges, I am curious about the convergence/divergence of the following series:
$$
\...
- 51
0
votes
0
answers
30
views
Prove the summability.
I'm reading a paper on distributed optimization and need to use the supermartingale theorem using summability. How can I prove that:
$$\sum_{k=0}^{\infty}\alpha(k)\sum_{s=0}^{k-1}\lambda_1^{k-1-s}\...
3
votes
0
answers
17
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summation of a series on a lattice
I am a physics student. And I am working on some wave function on some lattice. One question I encountered in my study is studying the following summation
$$f(x)=\sum_{m,n \in Z}\cos((m-n)gx)\exp(-\...
- 31
1
vote
0
answers
23
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Is the $d$-dimensional harmonic series is proportional to the surface-area of a $d$-sphere?
TLDR:
How to Prove:
$$ \sum_{\sqrt{a_0^2 + ... a_k^2} \le n, (a_0 .. a_k) \ne (0)^{k+1} } \frac{1}{(a_0^2 + ... a_k^2)^{\frac{k}{2}}} = \frac{k\pi^{\frac{k}{2}}}{\Gamma(\frac{k}{2} + 1)} \ln(n) + O(1)...
- 17.4k
1
vote
1
answer
37
views
Shifting Index of Recursive Sequence
If I have a recursive sequence defined by:
$a_0 = 7$
$a_n = a_{n-1} + 3 + 2(n-1),$ for $n \geq 1$
How is this recursive sequence the same as the one above. Isn’t $n+1 \geq 2$?
$a_0 = 7$
$a_{n+1} = a_{...
- 129
-2
votes
0
answers
33
views
Limits of series expansion
I'm a high school student and I'm a little confused with limits. Everytime our teacher uses Taylor series expansion of trigonometric(such as sinx) or algebraic functions(like exp(x)), he cancels ...
- 7
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votes
0
answers
36
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Find $n^ 1 +n^2+\dots. + n^k$? [duplicate]
How do I find $n^ 1 +n^2+\dots. + n^k$?
I found a post that asks about $n ^ 1 + n ^ 2 +\dots+ n^{n - 1}$, but I only want until $n^k$, and I can't apply the answer of that post to fit in my use.
...
-2
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0
answers
19
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Puntual Convergence of a Series [closed]
Puntual Convergence of $\frac{\sin(x^{2n})}{1+nx^n}$?
I think that The Ap Domain is $\Bbb R \setminus \{ \pi/2 + k\pi\}$ but I am not sure how I can prove it
1
vote
2
answers
42
views
Application of Stolz-Cesaro Theorem to a Series
I was given as an exercise to show that if $0 < s < 1$ and $\beta \geq 0$ then
$$\lim_{n\to\infty} (1-s)n^\beta\sum_{k=1}^n\frac{s^{n-k}}{k^\beta} = 1.$$ Clearly $\beta = 0$ is a trivial case ...
- 90
4
votes
0
answers
66
views
Is there a closed form for the quadratic Euler Mascheroni Constant?
Short Version:
I am interested in computing (as a closed form) the limit if it does exist:
$$ \lim_{k \rightarrow \infty} \left[\sum_{a^2+b^2 \le k^2; (a,b) \ne 0} \frac{1}{a^2+b^2} - 2\pi\ln(k) \...
- 17.4k
1
vote
0
answers
75
views
What $n$ would make $ \gcd _{k\ge1 }\left\{\dbinom{m+(k-1)\cdot n}{k}\right\}=1 \ \forall m \in \mathbb{N}$
Construct the sequence $a_n$
$$ a_n =
\begin{cases}
0, & \gcd _{k\ge1 }\left\{\dbinom{m+(k-1)\cdot n}{k}\right\}=1 \ \forall m \in \mathbb{N} \\
\min{m }: \gcd _{k\ge1 }\left\{\dbinom{m+(k-1)\...
- 6,332
2
votes
1
answer
82
views
Find the value of this gigantic expression containing $e$ and $\sin$
Find the value of $A$ where
$$A=\sum_{n=1}^{\infty}\left(n\sin\left(\frac{\pi n}{2}\right)\left(e^x-1-\frac{x}{1!}-\frac{x^2}{2!}-\cdots-\frac{x^n}{n!}\right)\right)$$
By using the expansion of $e^x$,...
1
vote
0
answers
46
views
Is there a rule for using parentheses or brackets after the summation symbol to indicate what is included in the sum? [duplicate]
Using parentheses or brackets removes ambiguity but is it necessary?
- 19
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0
answers
35
views
Fibonacci related sum $\sum_{k=0}^\infty\frac1{F_{2k+1}+1}$ [duplicate]
Find the sum $$\sum_{k=0}^\infty\frac1{F_{2k+1}+1}$$ where $F_k$ is a term in the Fibonacci sequence. I tried using $F_n$ in terms of the golden ratio but it doesn't seem to work.
