All Questions
Tagged with summation algebra-precalculus
977
questions
2
votes
1
answer
403
views
Simplifying the alternating sum of n squares
This question is based on a curious problem from Donald Knuth's The Art of Computer Programming, exercise 7 to chapter 1.2.1. It's stated as the following:
Formulate and prove by induction a rule for ...
- 848
3
votes
1
answer
66
views
How can i simplify the following formula: $\sum\limits_{i,j=1}^{n}(t_{j}\land t_{i})$?
Consider the following time discretization $t_{0}=0< t_{1} < ... < t_{n} = T$ of $[0,T]$ where the time increments are equal in magnitude, i.e. $t_{j}-t_{j-1}=\delta$.
How can i simplify the ...
- 1,838
0
votes
2
answers
87
views
How to prove that $(\sum_{i=1}^n a_i)(\sum_{i=1}^n b_i)= \sum_{i,j} a_ib_j$? [closed]
How to prove that $(\sum_{i=1}^n a_i)(\sum_{i=1}^n b_i)= \sum_{i,j} a_ib_j$? Is there any way to visualize the sums on both sides.
user907912
1
vote
1
answer
196
views
I wish to solve exactly this formula involving sums and products
I was solving a physics exercise and I encountered this formula:
$$\left< n_l \right>=\left[1+\sum_{k\neq l} \left(e^{bN(l-k)}\frac{\prod_{j\neq l} (1-e^{b(l-j)})}{\prod_{j\neq k} (1-e^{b(k-j)})}...
- 56
-1
votes
1
answer
72
views
Is there a way of simplifying $ \sum_{k=2}^{n} ke^{-a(k-2)^2}$?
Quick question, is there a way of further simplifying this sum
$$
\sum_{k=2}^{n} ke^{-a(k-2)^2}
$$
where $a>0$?
- 3,435
0
votes
1
answer
48
views
Sum Simplification
For $q,k\in\mathbb{N}$ and $1\leq q\leq N$, is the following simplification
$$
f(q,k)=\sum_{j=1}^Ne^{-(2k\pi\text{i})jq/N}=N\sum_{i=1}^k\delta_{iq,N}
$$
correct? Here, $\delta_{i,j}$ is the Kronecker ...
- 3,435
2
votes
1
answer
60
views
Evaluating $\sum_{cyc} \frac{a^4}{(a-b)(a-c)}$, where $a=-\sqrt3+\sqrt5+\sqrt7$ , $b=\sqrt3-\sqrt5+\sqrt7$, $c=\sqrt3+\sqrt5-\sqrt7$
Let $a=-\sqrt{3}+\sqrt{5}+\sqrt{7}$ , $b=\sqrt{3}-\sqrt{5}+\sqrt{7}$, $c=\sqrt{3}+\sqrt{5}-\sqrt{7}$. Evaluate:
$$\sum_{cyc} \frac{a^4}{(a-b)(a-c)}$$
What I have tried so far is writing the ...
- 379
2
votes
2
answers
153
views
$\sum_{k=0}^\infty[\frac{n+2^k}{2^{k+1}}] = ?$ (IMO 1968)
For every $ n \in \mathbb{N} $ evaluate the sum $ \displaystyle \sum_{k=0}^\infty \left[ \dfrac{n+2^k}{2^{k+1}} \right]$ ($[x]$ denotes the greatest integer not exceeding $x$)
This was IMO 1968, 6th ...
- 336
0
votes
1
answer
59
views
$\sum_{n=0}^{\infty} {n \choose y} p^{y+1}(1-p)^{2n-y}$
I am stuck in finding the sum of $\sum_{n=0}^{\infty} {n \choose y} p^{y+1}(1-p)^{2n-y}$.
The sum looks quite similar to a negative binomial sum but I can't really find the exact form. Can anyone help?...
- 243
1
vote
1
answer
558
views
$\sum _{k=1}^n\:\left(\cos\left(\frac{2\cdot k\cdot \pi }{n}\right)-2\:+\:i\cdot \sin\left(\frac{2\cdot k\cdot \pi }{n}\right)\right)$
$\sum _{k=1}^n\:\left(\cos\left(\frac{2\cdot k\cdot \pi }{n}\right)-2\:+\:i\cdot \sin\left(\frac{2\cdot k\cdot \pi }{n}\right)\right)$
Normally the general factor is $a(n)=\cos\left(\frac{2k\pi }{n}\...
- 37
1
vote
0
answers
46
views
can anyone simplify this equation? $\sum_{n=2}^{\infty} \sum_{j=1}^{n}\left[\frac{n !}{j !(n-j) !}(-A)^{n-j}(-B)^{j}\right]$
Can anyone help me simplify the following equation? Any ideas appreicate. Thanks a lot!
$\sum_{n=2}^{\infty} \sum_{j=1}^{n}\left[\frac{n !}{j !(n-j) !}(-A)^{n-j}(-B)^{j}\right]$
with two variables A ...
- 31
1
vote
1
answer
353
views
What is the fallacy in writing $x^2$ as the sum of $x$ $x's$? [duplicate]
It seems reasonable to write $x^2=x+x+...+x$ ($x$ times) but we run into a problem with derivatives if we do this.
The derivative of $x^2$ is $2x$ but the derivative of the sum on the right hand side ...
- 15
2
votes
2
answers
96
views
Simplifying a binomial sum
I came across the following expression
$$
-1+(1+a^5)x+4a^4bx^{h+1}+6a^3b^2x^{2h+1}+4a^2b^3x^{3h+1}+ab^4x^{4h+1}=0
$$
which I would like to simplify. I though of using a sum with the binomial ...
- 3,435
2
votes
1
answer
266
views
What is the difference between the integral$\int\limits_{}^{}$ and segments $\sum_{} ^{} $
I want to know, what is the difference between the integral and segments
I know that، Integration came when scientists asked how can calcule the area of unusual shapes، and one of they scientists came ...
- 39
1
vote
1
answer
47
views
Is a sum with two variables equal to two separate sums with one variable each? [closed]
Does $\sum_{i \in M, k \in S} f(i, k) = \sum_{i \in M} \sum_{k \in S} f(i, k)$ for all $f$?
- 135
0
votes
1
answer
113
views
Clean way to prove that $\sum_{k=0}^{N-1}\cos \left( \frac{2m \pi}{N}k\right)=0$
Here's the identity:
For $0 < m < N$, we have
$$
\sum_{k=0}^{N-1}\cos \left( \frac{2m \pi}{N}k\right) = 0
$$
I know I can solve this using a variety of methods, i.e. anything described here: ...
- 175
4
votes
1
answer
108
views
Clean proof for trigonometry identity? I know what the answer is, but I feel like there should be like a $1$-$2$ liner to compute this
Fix $j,k$ with $0 \leq j,k \leq N$. If $j+k$ is even, (i.e. if $j,k$ have same parity), then
$$
\sum_{n=1}^{N-1} \cos\left(\frac{j\pi}Nn\right)\sin\left(\frac{k \pi}Nn\right) = 0
$$
and
$$
\sum_{n=0}^...
- 175
0
votes
0
answers
148
views
How can I solve this two-sided infinite summation?
In this question, the comment suggests that the imaginary terms in the stated solution might sum to $0$. In order for this to happen, it must be the case that
$\sum_{-\infty}^\infty [\frac{2(-1)^{n + ...
- 1,922
2
votes
1
answer
1k
views
How to calculate sum of a sigma notation for sum expression with exponents?
So I have got this practice problem
$$\sum_{k=1}^n \frac{(-1)^k\cdot2^{2k}}{3}$$
Now I am fairly new to the sigma notation for the sum of elements. I know the basic stuff like the sum of the ...
- 53
6
votes
1
answer
105
views
calcuate $\sum_{i=0}^{n} 2^{2i}$
I want to calcuate this problem: $\sum_{i=0}^{n} 2^{2i+5}$
I know that we can expand this problem like this:
$\sum_{i=0}^{n} (2^{2i+5})$
$=\sum_{i=0}^{n} (2^5 \times 2^{2i})$
$=\sum_{i=0}^{n} (32 \...
- 337
0
votes
2
answers
96
views
Why does $x^n = \left[(x-1)\sum\limits_{k=0}^{n-1}x^k\right]+1$?
I was curious about the formula for the difference between a number powered by a certain number and the sum of the number powered from 0 to the number - 1. I found this formula but I'm not sure if ...
- 3
2
votes
1
answer
64
views
Summing minimums with floor function
Prove that for any positive integers $x, m, n$:
$$\sum_{i=1}^n\min\left(\left\lfloor\frac{x}{i} \right\rfloor,m\right)=\sum_{i=1}^m\min\left(\left\lfloor\frac{x}{i}\right\rfloor,n\right)$$
Intuitively ...
user949026
5
votes
1
answer
189
views
Evaluating $\sum_{r=1}^{89} \frac{1}{1+\tan^3 r}$
$$\sum_{r=1}^{89} \frac{1}{1+\tan^3 r }$$
where $r$ is in degrees
I tried this a lot using the $a^3+b^3$ identity but I don't seem to be getting anything fruitful :(
Can someone please give me a hint?...
- 205
0
votes
1
answer
65
views
Sigma summation to non-general summation
When doing definite integral as the limit of a sum I noticed they had changed Sigma summation to general summation not as general equation.
Here's an example
$$\lim_{h\rightarrow 0} h \sum_{r=1}^n (a+...
user953847
2
votes
2
answers
83
views
How was the closed form of $\sum_{j=i+1}^{n}(n-i-j+2)$ calculated?
I am struggling to understand the closed form calculation of the following triple summation:
$$
\sum_{i=1}^{n} \sum_{j=i+1}^{n} \sum_{k=i+j-1}^{n} 1
$$
The first step I understand:
$$
\sum_{i=1}^{n} \...
- 333
1
vote
2
answers
83
views
Why does $\sum_{j=i+1}^{n}(n-i-j+1) = n^{2}-5 n-2-i^{2}-3i$
Reading through https://stackoverflow.com/q/40696784/15753188 , the following calculation is performed
I see the rightmost summation is summing $(n-i-j+1)$, $n-(i+1)$ times. My first thought is to ...
- 333
0
votes
4
answers
114
views
Find the sum of the first $50$ terms of the series $a_{n} = -4a_{n-1} + 3$.
I'm not sure where to start this without being given some terms.
Find the sum of the first $50$ terms of the series $$a_{n} = -4a_{n-1} + 3$$ I can see that the common difference is $-4$ and the slope ...
- 172
2
votes
5
answers
126
views
How does $\sum_{k=j}^{i+j}(i+j-k)$ = $\sum_{k=1}^{i}(k)$
I am working with summations and I came across these two equivalent summations
$\sum_{k=j}^{i+j}(i+j-k)$ = $\sum_{k=1}^{i}(k)$ but there is no explanation as to how the latter summation was computed ...
- 333
0
votes
1
answer
87
views
Help required in finding Prakash's house number!!
The house number in Prakash's lane starts with natural numbers, i.e. $1,2,3, \ldots$. The sum of all the house numbers on the left side of Prakash's house is equal to the sum of all the house on its ...
1
vote
2
answers
124
views
How do I compute $\;\sum_{i=1}^{n} \sum_{j=1}^{i} \sum_{k=j}^{i+j} 1\;$?
I am working through the Algorithm Design Manual. Chapter 2 Problem 2 has us evaluate the return value of a function which consists of three nested loops which can be expressed by the following triply ...
- 333