All Questions
Tagged with summation algebra-precalculus
977
questions
-1
votes
0
answers
54
views
Summation involving the closest integer to $\sqrt n$ [closed]
Let $f(n)$ be the integer closest to $\sqrt n$. Evaluate
$$\sum_{n=1}^\infty\frac{\left(\frac32\right)^{f(n)}+\left(\frac32\right)^{-f(n)}}{\left(\frac32\right)^n}$$
In this question, I was able to ...
- 9
2
votes
0
answers
102
views
$x_{1}=\frac{1}{x_{1}}+x_{2}=\frac{1}{x_{2}}+x_{3}=\ldots=\frac{1}{x_{n-1}}+x_{n}=\frac{1}{x_{n}}$; prove that $x_{1}=2\cos\frac{\pi}{n+2}$ [closed]
If $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ are $n\geq 2$ positive real numbers such that $
x_{1}=\frac{1}{x_{1}}+x_{2}=\frac{1}{x_{2}}+x_{3}=\ldots=\frac{1}{x_{n-1}}+x_{n}=\frac{1}{x_{n}}$, prove that $...
- 1,122
2
votes
1
answer
82
views
If $M:=\sum\limits_{k=1}^{\frac{n(n+1)}{2}}\lfloor\sqrt{2k}\rfloor$ How to Find $\frac{n^3+2n}{M}$?
$$M:=\sum\limits_{k=1}^{\frac{n(n+1)}{2}}\lfloor\sqrt{2k}\rfloor$$
Find $\frac{n^3+2n}{M}$
This problem was on a problem book.
It is easy to find $M$
If $n$ is odd, $\ m=\frac{n+1}{2} $ and $$M= \...
- 3,338
18
votes
12
answers
17k
views
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
I am trying to prove this binomial identity $\displaystyle\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$ but am not able to think something except induction,which is of-course not necessary (I think) here,...
- 2,973
1
vote
0
answers
48
views
Is there a rule for using parentheses or brackets after the summation symbol to indicate what is included in the sum? [duplicate]
Using parentheses or brackets removes ambiguity but is it necessary?
- 19
1
vote
3
answers
93
views
Evaluating $\sum_{\substack{i+j+k=n \\ 0\leq i,j,k\leq n}} 1$
I need to find the sum $$\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,...
- 397k
2
votes
1
answer
104
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
$$\sum_{a <n <b} n^{-\frac{1}{2}-it} \ll (a/t)^{...
- 92.6k
1
vote
2
answers
187
views
summing binomial coefficiens related
$$
\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\ ?.
$$
One of my student had this question in his exam....
- 47.4k
2
votes
2
answers
398
views
Find the value of $S_1+S_2$
Knowing that $$\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
15
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 ...
- 3,845
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 ...
- 86
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