All Questions
Tagged with summation algebra-precalculus
974
questions
1
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46
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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
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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: $(1,0,0),(0,1,0), (0,0,1)$ $$\sum_{\substack{i+j+k=1 \\ 0\leq i,j,...
- 840
2
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1
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94
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Enquiry on a claim in Titchmarsh. [closed]
There is a claim on p.107 of Titchmarsh's ''The theory of the Riemann zeta function'', that if $0<a<b \leq 2a$ and $t>0$, then
the bound
$$\sum_{a <n <b} n^{-\frac{1}{2}-it} \ll (a/t)^{...
- 31
-1
votes
1
answer
55
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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 $a$.
0
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1
answer
14
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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
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1
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54
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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
3
votes
4
answers
82
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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 $$\sum_{n=1}^k\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}}=\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\dots +\...
- 524
1
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1
answer
80
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Sum related to Binomial Coefficients [duplicate]
Calculate:- $$\sum_{r=1}^{2023} \frac{(-1)^{r-1}r}{2024 \choose r}$$
And generalise the result if possible.
I've tried to reduce this to a telescopic sum but could not do it.
I've also made a ...
- 19
0
votes
1
answer
53
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On a strange step in the proof regarding a maximal problem.
As far as I'm concerned to show that something is true, proving that something is true for an example is never enough, you have to be able to prove that it is true for all statements/numbers with the ...
- 127
0
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0
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39
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Simplification of the ratio between series
I have been trying to solve a problem i posed to myself in the applied sciences, and technically, i did (though it is not of any practical use). But the problem is that the solution is, well, not ...
- 1
1
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1
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121
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$(an)_{n\in\mathbb{N}^*}$ is a sequence, so that $a_1=1$, for all $n\geq 2$, $a_n=a_1\cdot...\cdot a_{n-1}+1$. $\sum_{n=1}^{m}\frac{1}{a_n}<M$, find M
This is a problem I came across on another exam. It is as follows:
Let $(an)_{n\in\mathbb{N}^*}$ a sequence such that $a_1=1$ and for all $n\geq 2$, $a_n=a_1\cdot...\cdot a_{n-1}+1$. Find the real ...
- 61
1
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2
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187
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summing binomial coefficiens related
$$
\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\ ?.
$$
One of my student had this question in his exam....
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3
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87
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Evaluate using combinatorial argument or otherwise :$\sum_{i=0}^{n-1}\sum_{j=i+1}^{n}\left(j\binom{n}{i}+i\binom{n}{j}\right)$
Evaluate using combinatorial argument or otherwise $$\sum_{i=0}^{n-1}\sum_{j=i+1}^{n}\left(j\binom{n}{i}+i\binom{n}{j}\right)$$
My Attempt
By plugging in values of $i=0,1,2,3$ I could observe that ...
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0
votes
0
answers
44
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Inequality with Products and Sums
I need help to find a proof for the following inquality.
Assuming that $ 0 \leq c_i \leq 1 $ and $ 0 \leq d_i \leq 1 $, show that
$$
\prod_{i=1}^N (c_i + d_i - c_i d_i) \geq \prod_{i=1}^N c_i + \prod_{...
- 778
-3
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3
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106
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Why is $\sum_{n=1}^{k}\frac{1}{n^2+n}=\frac{k}{k+1}$ [duplicate]
$$\sum_{n=1}^{k}\frac{1}{n^2+n}=\frac{k}{k+1}$$
I don't think that this summation requires too much context as this is a Q&A site, but I was just wondering why the summation is evaluated so nicely....
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