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Results tagged with sequences-and-series
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user 62673
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
2
votes
0
answers
83
views
Convergence of a sequence involving a truncated exponential
Let $n\in\mathbb{N}_{>0}$, $\gamma\in\mathbb{R}_{>0}$. Let $\{a_n\}_{n}$ and $\{b_n\}_{n}$ be two sequences defined as follows
$$
a_n := \sum_{k=0}^{n-1}{2k \choose k}\frac{1}{\gamma^{2k+1}} \left(1-e …
- 2,712
6
votes
1
answer
373
views
Asymptotic behavior of a certain trigonometric partial sum
Let $a<0$ and $b>0$ be real numbers such that $a<-2b$. Let $n>1$ be a positive integer and consider the following partial sum:
$$
f(n) = \frac{1}{(n+1)^2}\sum_{i=1}^{n}\sum_{j=1}^{n} (-1)^{i+j}\frac{( …
- 2,712
3
votes
1
answer
316
views
Bounding a series of nested integrals
Consider the following matrix function
$$
f(t) = \cos(\omega_1t) A_1 + \cos(\omega_2t) A_2, \quad t\ge 0,
$$
where $A_1$, $A_2$ are real square matrices and $\omega_1$, $\omega_2$ positive numbers.
N …
- 2,712
4
votes
0
answers
82
views
Non-singularity of a series of matrices
Let $A_1$, $A_2$ be $n\times n$ real matrices. Suppose that $A_1$ and $A_2$ are Schur stable (i.e., their eigenvalues are strictly inside the unit circle in the complex plane). Let $B_1$, $B_2$ be two …
- 2,712