All Questions
Tagged with summation algebra-precalculus
975
questions
2
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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 ...
- 1,818
0
votes
0
answers
43
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Compute a tight upper bound of $\sum_{i=1}^{n-1}\frac{1}{3^i\log{n}- 3i}$?
I am trying to compute a tight upper bound of the sum below.
$\sum_{i=1}^{n-1}n\frac{\frac{1}{3^i}}{\log_3{(n/3^i)}}$
I was able to 'simplify' it up to the expression below.
$n\sum_{i=1}^{n-1}\frac{1}{...
- 185
0
votes
1
answer
86
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Can we derive the formula for summation of the series $1^{m}+2^{m}+3^{m}+...+n^{m}$ for every $m\geq2$? [duplicate]
Can we derive the formula for summation of the series $1^{m}+2^{m}+3^{m}+...+n^{m}$ for every $m\geq1$ ? For e.g. $1+2+3+...+n=\frac{n(n+1)}{2}$ . Here $m=1$ . Now I also know that $1^{2}+2^{2}+3^{2}+....
- 1,309
3
votes
1
answer
205
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Is there a sequence where the average of the squares is proportional to the square of the average?
Usually in math, the order of operations matters when you do anything more complicated than addition or multiplication. If you have a list of numbers, whether you apply the average or the squaring ...
- 103
0
votes
1
answer
81
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How to find the sum of a finite number of increasing fractional powers
It is easy to show
$\quad\sum_{i=1}^{\infty} \dfrac{1}{n^i}=\dfrac{1}{n-1}\quad$
e.g.
$$\quad\sum_{i=1}^{\infty} \dfrac{1}{2^i}=\dfrac{1}{1}\quad$ $\quad\sum_{i=1}^{\infty} \dfrac{1}{3^i}=\dfrac{1}{2}...
- 6,381
-1
votes
2
answers
241
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How to derive $1^3 + 2^3 + \cdots + n^3 = \left(\frac12n(n +1)\right)^2$? [duplicate]
So I know of the basic summation:
$$1 + 2 + \dots + n = \frac{n(n +1)}{2}$$
You derive this by noting that if you pair every element with the one on the other end (example: $1$ with $n$, $2$ with $n - ...
- 782
0
votes
0
answers
52
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How can I prove that $ \sum_{k=1}^{n}\frac{k\cdot P(n,k)}{n^{k+1}} = 1$? [duplicate]
The answer is difficult to me, I cannot figure out how to compute it.
$\sum_{k=1}^{n}\frac{k\cdot P(n,k)}{n^{k+1}}=1$
If someone can explain some technique to do it, I'd appreciate it. I tried to ...
- 1
0
votes
0
answers
104
views
series based on $(1+x+x^2)^n$
Question : Let $a_r$ denote the following $$(1+x+x^2)^n=\sum_{r=0}^{2n}a_rx^r$$
then prove the following
$$\sum_{r=0}^{n}(-1)^r\binom n r a_r = \begin{cases}
0 & n \ne 3k \text{ for all ...
- 931
8
votes
1
answer
250
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Compute $\sum^{2024}_{k=1} \frac{2023-2022 \cos \left(\frac{\pi(2k-1)}{2024} \right)}{2021-2020 \cos \left(\frac{\pi(2k-1)}{2024} \right)}$
Question: Compute $$\sum^{2024}_{k=1} \frac{2023-2022 \cos \left(\frac{\pi(2k-1)}{2024} \right)}{2021-2020 \cos \left(\frac{\pi(2k-1)}{2024} \right)}$$
I began by rearranging the sum as follows:
$$\...
- 640
2
votes
0
answers
84
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Evaluating $\sum_{k=0}^{\infty} \frac {2^k}{5^{2^k}+1}$ [duplicate]
So my teacher shared this problem with us and said everyone needs to try this, he teaches us Olympiad Math so I am assuming this wouldn't require analysis or calculus. This is the question, I have ...
- 51
2
votes
1
answer
72
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Proving $\sum_{k=0}^{(m-1)/2}(-1)^k{{m+1}\choose{k}}\left(\frac{m+1}{2}-k\right)^p=0$, for odd $m\geq3$ and even $2\leqslant p\leq m-1$
I am trying to prove the following identity for odd $m\geqslant 3$ and even $2\leqslant p\leqslant m-1$:
$$\sum_{k=0}^{(m-1)/2}(-1)^k{{m+1}\choose{k}}\left(\frac{m+1}{2}-k\right)^p=0$$
When I split ...
- 23
1
vote
2
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170
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Proof of weird formula for method of differences
Suppose we have to find sum of a sequence $t_1,t_2,t_3...t_n$.
For $1\le i\le n$, let $\triangle t_i=t_{i+1} -t_i$, $\triangle ^2t_i=\triangle t_{i+1}-\triangle t_i$ and so on ($\triangle ^{j}t_i=\...
- 433
0
votes
1
answer
56
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Prove the sum of n integers starting from n.
Prove that the sum of $n$ integers starting from $n$ has the recurrence $a_{n-1}+3n-2;\quad a_0=0$
$$\sum_{i=n}^{2n-1}i=a_{n-1}+3n-2\quad n\in\mathbb{N} \tag{EQ 1}$$
Examples:$$n=1 \qquad \sum_{i=1}^{...
- 1,211
1
vote
0
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92
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Example of a specific polynomial
I need four non-trivial polynomials $P(x)$, $Q(y)$, $R(z)$ and $S(w)$ such that $$P(x)Q(y)R(z)S(w)=\sum_{i}a_i b_i c_i d_i x^i y^i z^i w^i+ \sum_{j\neq k\neq l\neq m}e_j f_k g_l h_m x^j y^k z^l w^m $$ ...
- 862
1
vote
0
answers
47
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Multiple Sigma Notation and Expected Value
I've been struggling with expected value calculations that involve multiple or nested sigma notations. For instance, the solution to a problem I was working on is:
$X=\Sigma_{i=1}^{10}X_i\implies E[X]...
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