All Questions
Tagged with summation algebra-precalculus
974
questions
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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: $(1,0,0),(0,1,0), (0,0,1)$ $$\sum_{\substack{i+j+k=1 \\ 0\leq i,j,...
- 862
2
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1
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95
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Enquiry on a claim in Titchmarsh. [closed]
There is a claim on p.107 of Titchmarsh's ''The theory of the Riemann zeta function'', that if $0<a<b \leq 2a$ and $t>0$, then
the bound
$$\sum_{a <n <b} n^{-\frac{1}{2}-it} \ll (a/t)^{...
- 31
-1
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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
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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
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4
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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 $$\sum_{n=1}^k\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}}=\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\dots +\...
- 524
1
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1
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80
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Sum related to Binomial Coefficients [duplicate]
Calculate:- $$\sum_{r=1}^{2023} \frac{(-1)^{r-1}r}{2024 \choose r}$$
And generalise the result if possible.
I've tried to reduce this to a telescopic sum but could not do it.
I've also made a ...
- 19
0
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1
answer
53
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On a strange step in the proof regarding a maximal problem.
As far as I'm concerned to show that something is true, proving that something is true for an example is never enough, you have to be able to prove that it is true for all statements/numbers with the ...
- 127
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0
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39
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Simplification of the ratio between series
I have been trying to solve a problem i posed to myself in the applied sciences, and technically, i did (though it is not of any practical use). But the problem is that the solution is, well, not ...
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1
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1
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122
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$(an)_{n\in\mathbb{N}^*}$ is a sequence, so that $a_1=1$, for all $n\geq 2$, $a_n=a_1\cdot...\cdot a_{n-1}+1$. $\sum_{n=1}^{m}\frac{1}{a_n}<M$, find M
This is a problem I came across on another exam. It is as follows:
Let $(an)_{n\in\mathbb{N}^*}$ a sequence such that $a_1=1$ and for all $n\geq 2$, $a_n=a_1\cdot...\cdot a_{n-1}+1$. Find the real ...
- 61
1
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2
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187
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summing binomial coefficiens related
$$
\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\ ?.
$$
One of my student had this question in his exam....
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3
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Evaluate using combinatorial argument or otherwise :$\sum_{i=0}^{n-1}\sum_{j=i+1}^{n}\left(j\binom{n}{i}+i\binom{n}{j}\right)$
Evaluate using combinatorial argument or otherwise $$\sum_{i=0}^{n-1}\sum_{j=i+1}^{n}\left(j\binom{n}{i}+i\binom{n}{j}\right)$$
My Attempt
By plugging in values of $i=0,1,2,3$ I could observe that ...
- 9,569
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0
answers
44
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Inequality with Products and Sums
I need help to find a proof for the following inquality.
Assuming that $ 0 \leq c_i \leq 1 $ and $ 0 \leq d_i \leq 1 $, show that
$$
\prod_{i=1}^N (c_i + d_i - c_i d_i) \geq \prod_{i=1}^N c_i + \prod_{...
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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....
- 468
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4
answers
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Proving $\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$
I am trying to prove the following binomial identity:
$$\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$$
My idea was to use the identity
$$\binom{m}{m-n}=\binom{m}{n}=\sum_{i=0}^n(-...
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1
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3
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Attempt at creating a formula relating debt, payments and interest
I tried writing down a formula relating a given debt and interest to the periodic payments and number of payments.
So let's say someone starts off with a debt of $D$. The periodic interest is $r$ (for ...
- 1,037
3
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0
answers
35
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Prove that $\sum_{r=1}^n (-1)^{r-1}(1+\frac{1}{2}+\frac{1}{3}+...\frac{1}{r}) \binom{n}{r} =\frac{1}{n}$. [duplicate]
Prove that $\sum_{r=1}^n (-1)^{r-1}(1+\frac{1}{2}+\frac{1}{3}+...\frac{1}{r}) \binom{n}{r} =\frac{1}{n}$. Where $\binom{n}{r}$ represents 'n choose r'.
I tried to simplify this expression by first ...
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1
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41
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Conditions for $ \sum_{x\in G} f(x) = \sum_{x\in f(G)} x $ to hold
i have a question
what are the condition on the function $f$ ?
so that this equality hold :
$ \sum_{x\in G} f(x) = \sum_{x\in f(G)} x $
is $f$ surjective is a necessary for this question ?
please ...
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2
answers
82
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Upper rectangle area sum to approximate 1/x between $1\leq x\leq 3$
I am trying to figure out how to use rectangles to approximate the area under the curve $1/x$ on the interval $[1,3]$ using $n$ rectangle that covers the region under the curve as such.
Here is what I ...
- 1,071
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1
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Formula to increase x by y z times [closed]
What is the formula to increase x by y z times?
For example the number 4 I want to increase by 4 50 times. I added it out (4+8+12+16+20+24 etc....to 204) and got the right answer 5304 but what is ...
2
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4
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How did Rudin change the order of the double sum $\sum_{n=0}^\infty c_n\sum_{m=0}^n\binom nma^{n-m}(x-a)^m$?
I see many people change the order of sum but I don't understand how they did that.
Is there is a way to change the order of the sum, $\sum\limits_{k=a}^n\sum\limits_{j=b}^m X_{j,k}$ and $\sum\...
4
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3
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120
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Show $\sum_{i=0}^n{i\frac{{n \choose i}i!n(2n-1-i)!}{(2n)!}}=\frac{n}{n+1}$
How can this identity be proved?
$$\sum_{i=0}^n{i\frac{{n \choose i}i!n(2n-1-i)!}{(2n)!}}=\frac{n}{n+1}$$
I encountered this summation in a probability problem, which I was able to solve using ...
user1253495
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0
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98
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If $\sum_{i=1}^n x_i \ge a$, then what can we know about $\sum_{i=1}^n \frac{1}{x_i}$?
Suppose that $$\sum_{i=1}^n x_i \ge a$$
where $a>0$ and $x_i\in (0, b]$ for all $i$. Are there any bounding inequalities we can determine for $$\sum_{i=1}^n \frac{1}{x_i}?$$
I understand that $\...
- 1,615
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3
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I want to use integration for performing summation in Algebra
I am a class 9th student. Sorry if my problem is silly.
I am trying to find the sum of squares from 1 to 10. For this I tried summation, and it was fine.
But now I came to know that Integration can be ...
1
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0
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Simple algebra in rearring terms
I have a very simple mathematical question, and it is just about algebra which seems very tedious. First, let me state my problem from the beginning:
Let $i$ be an index representing countries ($i = {...
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0
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Finding a sufficient condition for dividends to be nonnegative
The Harsanyi dividend is defined as follows:
$d_v (S) = \sum_{R \subseteq S} (-1)^{|S|-|R|} v(R)$
Supermodularity is defined as follows, for $S \subseteq T \subseteq N$:
$v(S \cup \{i\}) - v(S) \leq v(...
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1
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Let $A_{k}=\{0,... ,n\}\setminus\{k\}.$ How to prove $\sum_{k=0}^{n}\left[(-1)^{k+1}\prod_{\substack{i,j\in A_{k}\\i<j}}(a_{i}-a_{j})\right]=0$?
Let $A_{k}=\{0,1,\ldots,n\}\setminus\{k\}$ for each $k=0,1,\ldots ,n$.
I think that the following equality is true for all $n\in\mathbb{N}, n\geq 2$ :
\begin{align}
\sum_{k=0}^{n}\left[(-1)^{k+1}\...
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1
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0
answers
98
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Restructuring Jacobi-Anger Expansion
In Jacobi-Anger expansion, $$e^{\iota z \sin(\theta)}$$ can be written as:
$$e^{\iota z \sin(\theta)} = \sum_{n=-\infty}^{\infty} J_n(z)e^{\iota n \theta}$$
where $J_n(z)$ is the Bessel function of ...
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2
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1
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Double Sum to Product Derivation
The function after the double-sigma sign can be separated into the
product of two terms, the first of which does not depend on $s$ and
the second of which does not depend on $r$. Source
Is the ...
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