All Questions
Tagged with summation algebra-precalculus
974
questions
18
votes
12
answers
17k
views
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
I am trying to prove this binomial identity $\displaystyle\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$ but am not able to think something except induction,which is of-course not necessary (I think) here,...
- 2,941
1
vote
0
answers
47
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
1
vote
3
answers
92
views
Evaluating $\sum_{\substack{i+j+k=n \\ 0\leq i,j,k\leq n}} 1$
I need to find the sum $$\sum_{\substack{i+j+k=n \\ 0\leq i,j,k\leq n}} 1$$
For $n=1$ we have the admissible values of $(i,j,k)$ as: $(1,0,0),(0,1,0), (0,0,1)$ $$\sum_{\substack{i+j+k=1 \\ 0\leq i,j,...
- 396k
2
votes
1
answer
95
views
Enquiry on a claim in Titchmarsh. [closed]
There is a claim on p.107 of Titchmarsh's ''The theory of the Riemann zeta function'', that if $0<a<b \leq 2a$ and $t>0$, then
the bound
$$\sum_{a <n <b} n^{-\frac{1}{2}-it} \ll (a/t)^{...
- 92.4k
1
vote
2
answers
187
views
summing binomial coefficiens related
$$
\mbox{If}\quad
s_{n} = \sum_{k = 0}^{n}\left(-4\right)^{k}
\binom{n + k}{2k},\quad\mbox{how to prove}\quad s_{n + 1} + 2s_{n} + s_{n - 1} = 0\ ?.
$$
One of my student had this question in his exam....
- 47.3k
2
votes
2
answers
381
views
Find the value of $S_1+S_2$
Knowing that $$\sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi^2}{6}$$
and $$S_i=\sum_{k=1}^{\infty}\frac{i}{(36k^2-1)^i}$$
Find value of $S_1+S_2$
i tried splitting:
$$\frac{1}{36k^2-1}=\frac{1}{2}\...
- 11
-1
votes
1
answer
55
views
Expressing $\sum_{b=0}^a\sum_{c=0}^b c$ in terms of $a$ [closed]
Summation with the form:
$$\sum_{b=0}^a\sum_{c=0}^b c$$
I am not aware of any rule about chaining sums and getting a value in terms of the variable $a$.
- 433
0
votes
1
answer
14
views
Solve for variable f when f is in a denominator function of a sum
I have the following equation which I need to solve for f:
$\frac{X}{fY} = \sum_t^{T-1}\frac{A(t)}{f\cdot B(t)+1}$
While this seems very solvable, it has stumped an entire group of physics students. ...
- 125
8
votes
3
answers
565
views
Finding sum : ${\mathop{\sum\sum\sum\sum}_{0\le i\lt j\lt k\lt l\le n }} \,1$
Finding the value of: $${\mathop{\sum\sum\sum\sum}_{0\le i\lt j\lt k\lt l\le n }} 1$$
I know a similar question was asked on this site earlier, but I couldn't understand the method used there.
Link ...
- 3,835
3
votes
2
answers
117
views
$1\binom{20}1+2\binom{20}2+3\binom{20}3+\dots+19\binom{20}{19}+20\binom{20}{20}$
$$1\binom{20}1+2\binom{20}2+3\binom{20}3+\dots+19\binom{20}{19}+20\binom{20}{20}$$
I solved it by letting the sum be $S$, then adding the sum to itself but taking the terms from last to first and then ...
- 76
0
votes
1
answer
54
views
Why is $\sum_{m=0}^{\lfloor xs\rfloor} 2 \binom{s}{m} p^m (1-p)^{s-m} \leq 2\exp{\left(-\frac{2(\lfloor xs\rfloor - sp)^2}{s}\right)}$
I am trying to understand few of the mathematical steps I have encountered in a paper, there are two of them
(a) $\sum_{m=0}^{\lfloor xs\rfloor} 2 \binom{s}{m} p^m (1-p)^{s-m} \leq 2\exp{\left(-\frac{...
- 141
1
vote
1
answer
80
views
Sum related to Binomial Coefficients [duplicate]
Calculate:- $$\sum_{r=1}^{2023} \frac{(-1)^{r-1}r}{2024 \choose r}$$
And generalise the result if possible.
I've tried to reduce this to a telescopic sum but could not do it.
I've also made a ...
- 19
3
votes
4
answers
82
views
Finding and proofing a closed formula for $\sum_{n=1}^k\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}}$
I want to find and proof a closed formula for the following sum $$\sum_{n=1}^k\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}}=\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\dots +\...
- 276k
0
votes
1
answer
53
views
On a strange step in the proof regarding a maximal problem.
As far as I'm concerned to show that something is true, proving that something is true for an example is never enough, you have to be able to prove that it is true for all statements/numbers with the ...
- 289
0
votes
0
answers
39
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Simplification of the ratio between series
I have been trying to solve a problem i posed to myself in the applied sciences, and technically, i did (though it is not of any practical use). But the problem is that the solution is, well, not ...
- 1