Tagged Questions
2
votes
0answers
38 views
Sum of squared/cube combinations [duplicate]
I was wondering if there is a closed formula for sum of cubed combinations. More precisely, I'd like to compute $$\sum_{k=1}^n \left ( \begin{array}{c}n\\k\end{array}\right )^3$$
Obviously, without ...
10
votes
3answers
142 views
Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
Prove
$$\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$$
I can't find counting interpretations for either of the sides. A hint of "if $S$ is a subset of $\{1, . . . , n\}$ and $S^\prime$ is its complement ...
0
votes
1answer
14 views
Clues to prove average in T is minor or equal than average in a smaller inner interval.
Suppose I want to prove (or disprove) this assertion
Let $f$ be a discrete function, $T,h,k$ are constants
So these terms are averages over $T$ and over $h$
$\sum\limits_{i=0}^{T}\frac {f(i)}{T}$ ...
3
votes
1answer
54 views
How to prove the identity $(n-k)! \sum _{i=0}^{n-k} \frac{(k+i-1)!}{i!} = \frac{n!}{k}$?
I am stuck in proving the following :
$$(n-k)! \sum _{i=0}^{n-k} \frac{(k+i-1)!}{i!} = \frac{n!}{k}$$
NOTE: I don't want any combinatorial proof. I think it is some algebraic manipulation.
0
votes
4answers
38 views
Summation of n-squared, cubed, etc. [duplicate]
How do you in general derive a formula for summation of n-squared, n-cubed, etc...? Clear explanation with reference would be great.
0
votes
1answer
24 views
Sum of series with generic term inside it
I have the following series:
$$
\sum_{k=0}^{+\infty} k \cdot a^k \cdot s_k
$$
Having $|a| < 1$ and where $s_k \in [0,1]$ is a generic sequence having the property for which $\lim_{k \to ...
0
votes
1answer
40 views
Help with understanding a summation formula
I am having trouble understanding the derivation of the summation formula below.
$$\sum_{k=1}^N \dfrac1{(k+1)(k+2)} = \dfrac{N}{2N+4}$$
4
votes
2answers
135 views
Compact form of sum (binomial coefficients)
Find compact formula of the following sum:
$$ \sum_{i,j,k \in \Bbb Z} {{n}\choose{i+j}}{{n}\choose{j+k}}{{n}\choose{k+i}} $$
Could you give me any HINT how to start it?
I've tried this way:
$$ ...
0
votes
1answer
57 views
Calculating a recursive power term binomial sum
Could someone please help me or give me a hint on how to calculate this sum:
$$\sum_{k=0}^n \binom{n}{k}(-1)^{n-k}(x-2(k+1))^n.$$
I have been trying for a few hours now and I start thinking it may ...
1
vote
2answers
93 views
Double summation, index change clarification.
As my teacher is not really helpful and he's just writing on the blackboard and not explaining what he's doing I have a question how did he obtain this : $$ ... = ...
1
vote
1answer
73 views
Iteration of the function $d(n)=a-n$
I start by defining the function $f$
$$f(0)=0,~~~~~f(n+1)=d(f(n))=a-f(n)$$
So:
$$f(n)=d^{\circ n}(0)$$
$f(1)=a-0=a$
$f(2)=a-(a-0)=0$
$f(3)=a-(a-(a-0))=a$
How can I find the solution of $f(x), ...
8
votes
2answers
149 views
Solve a summation
Hi guys I have an exercise I don't know how to approach, would be cool if you could give me a tip or two! A sequence $a_{n}$ is defined by a dependency : $$ \sum_{i, j, k \geq 0}^{i+j+k = n } ...
4
votes
2answers
70 views
$\sum_{i=1}^n i\cdot i! = (n+1)!-1$ By Induction
I am trying to prove the following by Mathematical Induction:
$$\sum_{i=1}^n i\cdot i! = (n+1)!-1\quad\text{for all integers $n\ge 1$}$$
My proof by Induction follows:
First prove $P(1)$ is true,
...
0
votes
1answer
33 views
Converting recurrence relation to summation, trivial problem
I am reading book concrete maths, in which at some point authors speak about general method of converting recurrence relation of type $a_n T_n = b_n T_{n-1} + c_n$ .
Then one multiplies the above ...
2
votes
0answers
39 views
How much is $\sum_{n\neq 0, n\in Z^d} \frac{\cos(n\cdot x)}{|n|^s}$
When $s\in (0,d)$ and $x\in [-\pi, \pi]^d$, how much is this sum :
$$\sum_{\large n\neq 0, n\in Z^d} \frac{\cos(n\cdot x)}{|n|\large^s}$$
If it does not converge, can I define it in some sort of ...
4
votes
3answers
123 views
Proving that $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots + \frac{1}{n(n+1)} = \frac{n}{n+1}$
How would we go about proving that $$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots +\frac{1}{n(n+1)} = \frac{n}{n+1}$$
2
votes
3answers
202 views
Sum of reciprocals of factorials
Could you help me count this sum:
$$ \sum_{n=1}^{9} \frac{1}{n!} $$
I don't think I can use binomial coefficients.
0
votes
1answer
88 views
Finite sum of products of binomial coefficients and quadratic polynomial
How can I calculate the value of such a sum?
$\sum_{k=0}^{n} (2k^2-3k+1){n\choose k}$
Should I split it into three sums? But then I don't know what to do with $k^2{n\choose k}$. I know that ...
3
votes
1answer
219 views
Proof by induction for a summation?
Looking for some help with a proof by iduction. Im looking to proove the following summation holds true:
$$\frac{\langle W \vert (C)^{N} \vert V \rangle}{\langle W \vert \vert V ...
0
votes
1answer
61 views
Why is the upper bound of this statement always incremented by 1?
Why is "for j = 1 to n" translate to this? Why is the upper bound always incremented by 1?
$$\sum_{j=1}^{n+1}1 $$
Why isn't it
$$\sum_{j=1}^n1 $$
"for j = 1 to n" is written in pseudo code btw ...
1
vote
2answers
142 views
Solving the recurrence relation $p(n,m) = n \times \sum\limits_{k=n-1}^{m-1} p(n-1,k)$ where $p(1,m) = m$
I am trying to solve the following recurrence relation
$$p(n,m) = n \times \sum\limits_{k=n-1}^{m-1} p(n-1,k)$$
$$p(1,m) = m$$ $$p(0,0)=0$$
Any hints or ideas?
(Not a homework assignment)
Edit: n ...
5
votes
1answer
68 views
Proving at least 99 pairwise sums of the reals are non-negative
Suppose, one hundred real numbers are given and their sum is 0.Then how can I prove that at least 99 of the pairwise sums of these hundred numbers are non-negative?
I tried this:
Let the real numbers ...
1
vote
2answers
211 views
How do I simplify nested summations? Also how does it work? Like a loop?
$$
\sum^{n}_{j=1}\sum^{n}_{k=1} jk
$$
How do I simplify this? Also can someone explain how this works?
its gonna be like this right? or am I misunderstanding it?
$$
\sum^{n}_{j=1}(j+2j+3j+...+nj)
$$
0
votes
2answers
48 views
Why doesn't this formula work in summation of cubes? but works in summation of squares
$$
\sum^{n}_{k=1} k^3 = ({n^2(n+1)^2})/4
$$
right?
say for example k not equal to 1, why doesn't this work? I subtracted the summation of k-1?
$$
\sum^{n}_{k!=1} k^3 = ({n^2(n+1)^2 - (k-1)^2k^2})/4
...
2
votes
2answers
191 views
Chain rule for discrete/finite calculus
In the context of discrete calculus, or calculus of finite differences, is there a theorem like the chain rule that can express the finite forward difference of a composition $∆(f\circ g)$ in ...
0
votes
2answers
421 views
How many distinct functions can be defined from set A to B
In my discrete mathematics class our notes say that between set A (having 6 elements) and set b (having 8 elements), there are 8^6 distinct functions that can be formed, in other words: |b|^|a| ...
