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Results tagged with sequences-and-series
Search options answers only
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user 11260
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.
79
votes
Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\ga...
The intuition may be helped by considering the generalized Euler constant function
$$\gamma(z)=\sum_{n=1}^\infty z^{n-1}\left(\frac{1}{n}-\ln\frac{n+1}{n}\right),\;\;|z|\leq 1.$$
Its values include t …
- 188k
31
votes
Function that produces primes
For the record (not an answer), the function $a(n-1,n)$ for $n$ up to $10^4$ contains 2264 distinct primes, the largest being equal to 20369. I checked that no primes are missing. The growth rate of t …
- 188k
31
votes
Fibonacci series captures Euler $e=2.718\dots$
Mathematica tells me it's a consequence of the two series (distinguished by $\pm$):
$$\sum_{k=0}^\infty\frac{F_{n\pm k}}{k!}=\frac{e^{\sqrt{5}} \phi^n-(1-\phi)^n}{\sqrt{5}\, \exp(\phi^{\mp 1})},$$
wit …
- 188k
20
votes
Accepted
Kindda-Perfect number: Is there a sequence of numbers which are equal to the sum of its prop...
Such an abundant number with abundance 1 is called a quasiperfect number (which is a more professional way to say "kindda-perfect"). None have been found, according to Wikipedia. This 1982 article say …
- 188k
16
votes
Series and sequences in physical systems & closed form expressions
The Casimir effect is a manifestation of
$$1+2^3+3^3+\cdots=-\frac{1}{120}.$$
The vacuum energy $E$ in the space between two metal plates, separated by a distance $a$ equals
$$E = \frac{ \hbar c \ …
15
votes
Accepted
Integrality of a binomial sum
$$\sum_{k=1}^n\frac{(4k - 1)4^{2k - 1}\binom{2n}n^2}{k^2\binom{2k}k^2}=16^n \left(1-\frac{\Gamma \left(n+\frac{1}{2}\right)^2}{\pi \Gamma (n+1)^2}\right)$$
$$\qquad=2^{4n}-c_n^2,\;\;\text{with}\;\;c_ …
- 188k
15
votes
Accepted
In search of an alternative proof of a series expansion for $\log 2$
Since you wish to develop techniques, you might want to consider the more general form
$$S_k=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^k\binom{2n}n2^n}.$$
The arcsine representation
$$\arcsin^2z=\frac12\ …
- 188k
15
votes
Accepted
Closed form for $\sum_{n=0}^\infty \frac1{2^{2^n}}$?
If you allow for a named number to be a closed form representation, the answer is "yes".
$\sum_{n=0}^\infty (1/2)^{2^n}$ is known as the Kempner number [1], a transcendental number [2].
More generally …
- 188k
14
votes
Accepted
Asymptotic behavior of $\sum_{k=1}^{\infty} \sqrt{\max\{1 - k^2/x^2,0\}}$ as $x\to\infty$
Use the Euler-MacLaurin formula,
$$\sum_{k=1}^\infty F(k)=\int_0^\infty F(k)\,dk+\tfrac{1}{2}[F(\infty)-F(0)]+\int_0^\infty (k-\text{Int}\,[k]-\tfrac{1}{2})F'(k)\,dk.$$
In this case $F(k)=\sqrt{\max\{ …
- 188k
14
votes
Behavior of $\sum_{n=1}^{\infty} (\{n \xi \} - \frac{1}{2})$ for irrational number $\xi$
This is studied in detail in Sums of Fractional Parts of Integer Multiples of an Irrational
The
sum
$$C(N)=\sum_{n=1}^{N} \bigl( \{ n \xi \} - \tfrac{1}{2}\bigr)$$
satisfies $|C(N)|>c\log N$ for infi …
- 188k
13
votes
Evaluating an infinite sum related to $\sinh$
this is not a proof – GH has given that – but I just want to note four more general series of this type listed in Andreas Dieckmann's extensive collection:
the OP's sum is the fourth series (or the …
- 188k
13
votes
Accepted
Are there any identities for alternating binomial sums of the form $\sum_{k=0}^{n} (-1)^{k}k...
A rewrite of formula (10) on MathWorld (replacing the summation index $k-i\mapsto i$) gives the desired formula:
$$\sum_{k=0}^{n} (-1)^{k}k^{p}{n \choose k} =(-1)^n n! S_2(p,n),$$
where $S_2(p,n)$ is …
- 188k
13
votes
Slick proof of Stirling's Formula?
The proof in the OP based on the sequence $a_n$ is proof number 1 in Steve Dunbar's Dozen Proofs of Stirling’s Formula (page 8, worked out here). Is there an alternative proof based on a sequence $b_n …
- 188k
12
votes
Accepted
turn $\pi/n$, move $1/n$ forward
The large-$n$ asymptotics of the harmonic numbers is $H_n\simeq \gamma_E+\log n$, which we use for $n\geq k_0$, replacing the sum $\sum_{n=k_{0}}^k$ by an integral $\int_{k_0}^k dn$. We thus find
$$\b …
- 188k
12
votes
Accepted
Closed form of an infinite series
Q: Does the following infinite series have a closed form?
It does, according to Mathematica:
$$\sum_{n=1}^{\infty} {(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin\left(\frac …
- 188k