All Questions
26
questions
1
vote
4
answers
3k
views
How to calculate this double sum? [closed]
This occurred in a probability problem where I have to calculate the invariant $c$ which equals to $N$ divided by the following double summation:
$$\sum_{n=0}^{N} \sum_{k=0}^N |k-n|$$
4
votes
6
answers
503
views
Proof of equation with binomial coefficients: $\sum\limits_{k=1}^{n} (k+1) \binom{n}{k} = 2^{n-1} \cdot (n+2)-1$ [duplicate]
$$\sum\limits_{k=1}^{n} (k+1) \binom{n}{k} = 2^{n-1} \cdot (n+2)-1$$
Maybe it's simple to prove this equation but I'm not sure how to get along with the induction. Any hints for this? Or may I use ...
- 824
3
votes
1
answer
88
views
Any simpler form for $ \frac{\sum_{k=2}^{n-2}{k\left(\sum_{i=0}^{k}\frac{(-1)^i}{i!}\right)}}{n\sum_{i=0}^{n}\frac{(-1)^i}{i!}}$
Is there any simpler form for the following expression:
$$
\frac{\sum_{k=2}^{n-2}{k\left(\sum_{i=0}^{k}\frac{(-1)^i}{i!}\right)}}{n\sum_{i=0}^{n}\frac{(-1)^i}{i!}}$$
Because I have to compute this ...
- 1,445
0
votes
4
answers
145
views
Any simpler expression for$\frac{\sum_{k=2}^{n-2}{k\big(\sum_{i=0}^{n-2}\frac{(-1)^i}{i!}\big)}}{n\sum_{i=0}^{n}\frac{(-1)^i}{i!}}$
Is there any simpler form for the following expression:
$$
\frac{\sum_{k=2}^{n-2}{k\left(\sum_{i=0}^{n-2}\frac{(-1)^i}{i!}\right)}}{n\sum_{i=0}^{n}\frac{(-1)^i}{i!}}$$
Because I have to compute this ...
- 1,445
0
votes
1
answer
31
views
Divergence of $ \sum_n\sqrt{2\pi}^{-1}{n^{-r^2/2}}\left(\frac{1}{r\sqrt{\log n}} - \frac{1}{r^3(\log n)^{3/2}}\right)$.
From this MathOverflow post, we have the following.
$$
\sum_n\sqrt{2\pi}^{-1}{n^{-r^2/2}}\left(\frac{1}{r\sqrt{\log n}} - \frac{1}{r^3(\log n)^{3/2}}\right)$$
This diverges if $r^2/2\le 1$, i.e., ...
user252435
3
votes
2
answers
275
views
$\sum_{k=n}^{\infty}\left(n-k\right)e^{-\lambda}\frac{\lambda^{k}}{k!}= ?$
Could you please help me. How do I sum the following:
$$\sum_{k=n}^{\infty}\left(n-k\right)e^{-\lambda}\frac{\lambda^{k}}{k!}$$
If the summation had started at 0, then it would be simply an ...
- 31
1
vote
1
answer
51
views
Summand Evaluation Help
I'm a student currently in an algorithms and data structures class, and my Calculus is unfortunately quite shaky when it comes to summations. As such, I'm struggling to evaluate one of the sum that ...
- 11
-1
votes
2
answers
2k
views
Help me understand how to take derivative of the PDF of X~binom(n,p) with respect to p.
This is the solution I was given.
My questions:
Why is it summed from k=1 to x. Shouldn't it be from k=1 to n? (If not, why not?)
What is happening to the first term from line 1 to line 2? When we ...
- 1,197
3
votes
1
answer
628
views
Is this infinite sum always less than zero?(+500pts bounty for the correct answer)
I wonder if the following infinite sum is always negative for all (finite) $A,d>0$ and $B<0$. Any counterexample also suffice. Here is the sum:
$$\frac{\partial}{\partial d}\sum_{n=1}^\infty n \...
- 7,892
0
votes
2
answers
600
views
What does this series converge to?
What does the following expression converge to?
$${\sum_{i = 1}^n{\left(\frac{S-s_i}{S}\right)^S}}$$
Where the sum of the $s_i$'s equals $S$.
How do you work out what it converges to?
1
vote
1
answer
311
views
Expected value of a Poisson sum of confluent hypergeometric functions
How to compute the expected value of a Poisson sum of the following confluent hypergeometric function:
$$
\sum_{y=1}^{Y} {}_1F_1(y,1,z)
$$
where y is positive integer taking values from the Poisson ...
- 510