Questions tagged [summation]

11,405 questions
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How to calculate the number of elements in all subsets of a set

There are $2^N$ subsets of a set. For instance the set $\{ 1, 2 \}$ has the following subsets: $\{ \}$ $\{ 2 \}$ $\{ 1 \}$ $\{ 1, 2 \}$ I'm trying to calculate the total number of elements in all of ...
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If $a,b,c>0$ and $ab+bc+ca=3$, prove that $\sum_{cyc} \frac{a}{\sqrt{a^3+5}} \leq \sqrt{6}/2$

If $a,b,c>0$ and $ab+bc+ca=3$, prove that $\displaystyle \sum_{cyc} \frac{a}{\sqrt{a^3+5}} \leq \sqrt{6}/2$. My attempt was to use firstly AM-GM in the denominator, like $a^3+5 \geq 3a+3$ and the ...
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Can Leibniz integral rule be extended to differentiation under the sigma sign?

To differentiate $\displaystyle M(t)=\sum_i e^{tx_i} P(x_i)$ with respect to $t$, for instance.
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Inequality relations used in Multiple Sums

I am reading Chapter 2 of Concrete Mathematics, and have some trouble understanding the rationale behind two identities used for simplification. ...
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How to show that if $\sum_{d|(p-1)} \psi(d) = \sum_{d|(p-1)} \phi (d)$ then $\psi(d) = \phi (d)$

The argument used in the book is that they will show that $\psi (d) \leq \phi(d)$ for each divisor $d$ of $p-1$ because this in conjunction with the equality of the two sums in the title will produce ...
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Solve and asymptotic expansion of $\sum_{a=1}^{H} \sum_{b=a+1}^{H} \left\lfloor{\frac{H}{a\, b}}\right\rfloor$

I am solving constrained polynomial systems resulting in constrained sums. I am looking to see if $$\sum_{a=1}^{H} \sum_{b=a+1}^{H} \left\lfloor{\frac{H}{a\, b}}\right\rfloor$$ is expressible in ...
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Prove that $\sum\limits_{x=0}^\infty \frac{1}{(x+ 1)(x+2)} = 1$.

Prove $$\sum_{x=0}^\infty \frac{1}{(x+ 1)(x+2)} = 1.$$ I couldn't find this problem solved online and I haven't reviewed series in a long time. I thought maybe squeeze theorem could help? A related ...
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Is there such an infinite sequence, such that $\lim_{n\to\infty} \frac{\sum_{i=1}^{n}a_n}{2n}=\text{ exact form constant}?$

I will try to ask my question as clear as possible. We know that, there exist infinitely number of infinite sequences that, consist of elements $\left\{0,1,2 \right\}$, which is can not express by ...
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Prove that $\sum_{k=0}^{2n} \frac{1}{\pi^n+\pi^k}={\frac{2n+1}{2\pi^n}}.$

Prove that $$\sum_{k=0}^{2n} \frac{1}{\pi^n+\pi^k}={\frac{2n+1}{2\pi^n}}.$$I tried telescoping taking $$t_k=\frac{1}{\pi^n+\pi^k}$$ but I am unable get the differencing: $t_k=f_k-f_{k-1}.$ Can some ...
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Compute difficult sum

How can I compute this sum? $\sum_{k=0}^\infty\left[\frac{(2k+1)!}{(2k+b)}-\frac{1}{(2k)!(2k+b)}-\frac{(2k)! }{(2k+b+1)}+\frac{1}{(2k+1)!(2k+b+1)}\right]$ I see the $k=0$ term is equal to zero, but ...
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Integral and Summation relationship [closed]

Summation $$\sum_{k=-\infty}^{+\infty}x_k$$ isn't discrete version concept of Integrals $$\int_{-\infty}^{+\infty} f(x) dx$$, that is obvious. Nevertheless I don't understand why when we come from ...
Find the value of $a_4-a_2$
Let $a_1<a_2<a_3<a_4$ be positive integers such that $\sum_{i=1}^{4}\frac{1}{a_i}=\frac{11}{6}$. Find the value of $a_4-a_2.$ I do not know how to proceed. I have tried to simplify the ...
What is $2\sum^2_{i=1}\sum^2_{j>i}a_{ij}x_ix_j$ is equal to?
Intuitively I did: $$2\sum^2_{i=1}\sum^2_{j>i}a_{ij}x_ix_j=2(a_{12}x_1x_2)+2(a_{22}x_2x_2)$$ But I'm not sure, since I'm new to calculus.