Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [summation]

Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

The notation $\sum\limits_{i=1}^na_i$ means $a_1+\ldots +a_n$.

Use for sums of infinite series and questions of convergence; use for questions about finite sums and simplification of expressions involving sums.

17770 questions
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A closed form for $1^{2}-2^{2}+3^{2}-4^{2}+ \cdots + (-1)^{n-1}n^{2}$

Please look at this expression: $$1^{2}-2^{2}+3^{2}-4^{2} + \cdots + (-1)^{n-1} n^{2}$$ I found this expression in a math book. It asks us to find a general formula for calculate it with $n$. The formula that book suggests is this:…
BiriBora
  • 71
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Confused about infinite sum $\sum\limits_{a,b,c}\frac{a+b+c+abc} {2^a(2^{a+b}+2^{b+c}+2^{a+c})}$

$$ \displaystyle \sum^{\infty}_{a=0} \displaystyle \sum^{\infty}_{b=0} \displaystyle \sum^{\infty}_{c=0}\dfrac{a+b+c+abc} {2^a(2^{a+b}+2^{b+c}+2^{a+c})}= \ ? \ $$ I calculated its value as $\frac{32}{3}$ but I'm not sure whether I'm right or…
Akshat Sharda
  • 117
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6 answers

Prove a lower bound for $\sum_{i=1}^n i^2$

Prove that $$\sum_{i=1}^n i^2 \geq \frac{n^3}{3}$$ for all $n \geq 1.$ What I know: I know the basic format of how to make a proof with the basis and inductive step but I am unsure of how to prove this particular statement and expand it. This is…
Connie
  • 131
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4 answers

Surprising Summation (2): $\frac14 \sum_{i=1}^{2n}i(2n-i+1)=\sum_{i=1}^n i^2$

Show that $$\frac14 \sum_{i=1}^{2n}i(2n-i+1)=\sum_{i=1}^n i^2$$ without expanding the summation to its closed-form solution, i.e. $\dfrac 16n(n+1)(2n+1)$ or equivalent. E.g., if $n=5$, then $$\frac 14 \bigg[\;1(10) + 2(9)+…
Hypergeometricx
  • 22,657
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Surprising Summation (1) $\sum_{i=1}^n(n-i+1)(2i-1)=\sum_{i=1}^n i^2$

Show that $$\sum_{i=1}^n(n-i+1)(2i-1)=\sum_{i=1}^n i^2$$ without expanding the summation to its closed-form solution, i.e. $\dfrac 16n(n+1)(2n+1)$ or equivalent. E.g., if $n=5$, then $$5(1) + 4(3)+ 3(5)+2(7)+1(9)=1^2+2^2+3^2+4^2+5^2$$ Background…
Hypergeometricx
  • 22,657
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2 answers

Sum of Squares in terms of Sum of Integers

We know that the sum of squares can be expressed as a multiple of the sum of integers as follows: $$\begin{align} \sum_{r=1}^n r^2 &=\frac 16 n(n+1)(2n+1)\\ &=\frac {2n+1}3\cdot \frac {n(n+1)}2\\ &=\frac {2n+1}3\sum_{r=1}^nr\end{align}$$ Is there a…
Hypergeometricx
  • 22,657
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2 answers

Find the sum of series $(1^2+1)1!+(2^2+1)2!+(3^2+1)3!+...+(n^2+1)n!$.

Find the sum of series $(1^2+1)1!+(2^2+1)2!+(3^2+1)3!+...+(n^2+1)n!$ I have found one method as i have shown in my answer below. But that form took me 30 mins to identify. $T_n=(n^2+1)n!$=$((n+1)(n+2)-3(n+1)+2)n!$ Hence adding all the terms and…
user227000
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5 answers

Is $\sum_ix_iy_i=\sum_ix_i\sum_iy_i$?

I ask this because the equation for the center of mass of a system (made up of a number of small masses attached to each other) is given by: $$\bar x=\frac{\sum_im_ix_i}{\sum_im_i}$$ If the operation in the question is valid then $\sum_ix_i$ would…
RobChem
  • 909
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How to simplify $\sum_{a_1=1}^n\sum_{a_2=1}^{a_1}\sum_{a_3=1}^{a_2}\dots\sum_{a_{k+1}=1}^{a_k}1$

Let $$x=\sum_{a_1=1}^n\sum_{a_2=1}^{a_1}\sum_{a_3=1}^{a_2}\dots\sum_{a_{k+1}=1}^{a_k}1$$ where $n,k\in\mathbb{Z}^+$. How to simplify $x$? I simplified it for $k=1,2,3$ and I got $n$, $\dfrac12n(n+1)$ and $\dfrac16n(n+1)(n+2)$. From this I assumed…
user164524
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2 answers

Summing a series to $n^{2}$

I'm currently trying to to sum the following series: $$ \sum_{k=1}^{n^2}\frac{1}{1 + \left ( \frac{k}{n} \right)^{r}} $$ I'm not sure what to do since we are summing over $n^{2}$. Any help would be appreciated.
Chaz
  • 45
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Calc the sum of $\sum_{k = 0}^{\infty} \frac{(-1)^k}{k} \sin(2k)$

Solving a bigger problem about Fourier series I'm faced with this sum: $$\sum_{k = 0}^{\infty} \frac{(-1)^k}{k} \sin(2k)$$ and I've no idea of how to approach this. I've used Leibniz convergence criterium to verify that the sum should have a value,…
iveqy
  • 1,327
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Why $\sum\limits_{i=1}^n \frac{1}{i} =\mathcal O(\ln(n))$?

I am reading textbook proof on algorithms that uses this fact: $$\sum_{i=1}^n \frac{1}{i} =\mathcal O(\ln(n)).$$ Why is that true? I'm trying to use Taylor series to find out why, but I'm not making much progress.
David Faux
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Solve for stopping point of sum

I am currently in 8th grade and in my Algebra class we are currently covering exponential growth, e.g. bacteria splitting, fruit fly growth, etc. Anyway, one exercise we did was on zombies. The problem started with 5 "sleeper cell" zombies, who each…
Milo
  • 347
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1 answer

How to calculate the integer part of the value of the following equation?

How to calculate the integer part of the value of the following equation? $$y=1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+\ldots+\frac{1}{\sqrt{1000000}}$$ It should be calculated in a special way, after all the equation is so long.
Railgun
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What is sum of the Bernoulli numbers?

$$\sum_{n=0}^\infty B_n = ?$$ I tried Wolfram Alpha but I can't seem to get the input correct.
zerosofthezeta
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