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
0
votes
1 answer

$\sum_\eta \exp(b*(\eta_1 + ...+\eta_n)) = (1+e^b)^n$.

I'm wondering why this is a correct relation, $\eta$ is the collection of $\{\eta_1,...,\eta_n \}$; ${\eta_1,...,\eta_n }$ are independent and the given relation is : $\sum_\eta \exp(b*(\eta_1 + ...+\eta_n))= \sum_{\eta_1} \sum_{\eta_2}…
Mari3
  • 95
  • 2
  • 10
0
votes
1 answer

Exact Hat Algebra

When computing changes in $\lambda_{ij}$ defined as $\hat \lambda_{ij} = {\lambda_{ij}^{'} \over \lambda_{ij}}$, where does the $\lambda_{lj}$ in the sum in the denominator come from? $$\lambda_{ij} = {\chi_i (\tau_{ij} w_i)^{-\epsilon} \over…
mfauth
  • 3
0
votes
2 answers

Consider the Sequence and prove.

Question: Consider the sequence defined as $a_1 = 2$ and $a_k = a_{k-1}+2k-1$ for all positive integer $ k \geq 2$ . Show that $a_n = 1+\sum(2i-1, i = 1 .. n)$ . Hint: Start with $\sum(2i-1, i = 1 .. n)$ and use the recursive definition…
Salan
  • 1
0
votes
0 answers

Finite sum of a floor function

Another question from an old MPhil paper: Is there a way to compute an expression for $$\sum_{y=1}^{\lfloor0.5(\sqrt{2x-1}-1)\rfloor}\lfloor0.5(\sqrt{x-y^2}-y-1)\rfloor$$ Other than brute force case-by-case sum I am unable to come up with an…
Wilbur
  • 61
0
votes
2 answers

Can this be simplified to below?$\frac{(\sum_{i=1}^{n} X_i Y_i)}{(\sum_{i=1}^{n} X_i^2)} = \frac{\sum_{i=1}^{n} Y_i}{\sum_{i=1}^{n} X_i}$

Can this be simplified to below? $$\frac{(\sum_{i=1}^{n} X_i Y_i)}{(\sum_{i=1}^{n} X_i^2)} = \frac{\sum_{i=1}^{n} Y_i}{\sum_{i=1}^{n} X_i}$$
Tree Garen
  • 709
  • 5
  • 15
0
votes
1 answer

Help me to understand the summation notation

I am having following expression, limits of summations are given in terms of set, can someone help me to interpret this summation. where $d_{1}$ and $d_{2}$ are distances. $$\sum_{i_1,i_2 \in \{1,2\}} d_{i_1} e^{-d_{i_1}} * d_{i_2} e^{-d_{i_2}}$$
M. Liaqat
  • 1
0
votes
2 answers

how to write $X_1 X_2 + X_2 X_3$ in the form of Summation?

i am trying to write this formula $$X_1 X_2 + X_2 X_3$$ in the form of summation. is this right? $$\sum_{i < j} X_i X_j, \quad i,j \in\{1,2,3\}$$ if not, please give a right one.
JJJohn
  • 1,436
0
votes
1 answer

summation dependent on previous term

In answering a question myself I have ended up with a summation of the form: $$a_n=\frac{n(n+3)}{2}a_{n-1}+\frac{n(n+1)}{2}$$ I am unsure of how to solve this type of summation such that I have it in the form: $$a_n=f(a_0,n)$$ I know that problems…
Henry Lee
  • 12,215
0
votes
1 answer

Deriving a Summation for a Series of Loops

I'm trying to solve a problem in which I look at a fragment of pseudocode and try to develop summations from it so I can analyze the time complexity of the overall loop. The code is this: for i=1 to n: for j=1 to n: if n<(n+i)/2: …
bpryan
  • 65
  • 1
  • 7
0
votes
1 answer

How to sum a multiset

I'd like to express the sum of every element in a multiset using sigma and/or universal quantification. Let A be a multiset. The function of x equals the sum of every element of A. x should equal 8. $$A = \{1, 2, 2, 3\}$$ Thank you,
WarmTaunTaun
  • 47
0
votes
1 answer

Unsure how the following summation simplifies down to this known result?

How does this: $$\frac{1}{c+1} + (c-1)\cdot\left(\frac{1}{c(c+2)}+\dotsb+\frac{1}{(n-2)n}\right) + \left(\frac{c-1}{n-1}\right)\cdot\frac{1}{2}$$ Become this: $$= \frac{2cn-c^2+c-n}{cn}$$ PS. $1 \leq c \leq n$. Not sure if that's needed or…
Penguin47
  • 57
0
votes
1 answer

Creating a formula for using energy while regaining it

Backstory: Im playing a game and I regain energy in this game a 1 / 180 s (3m). I would like to be efficient at using my energy, and use energy over a span of 8 hours without running out. At max, I can hold 130 energy in my storage for us, and I use…
Ice76
  • 101
0
votes
2 answers

What is the sum $\sum_{k=0}^{\infty} kz^{-k}$?

Any hint clarifying the problem as stated in the title, i.e what is $\sum_{k=0}^{\infty} kz^{-k}$? would be very appreciated.
WorseThanEinstein
  • 187
0
votes
1 answer

What is the sum $\sum_{n=n_0-N+l}^{n_0+N}1$?

To be very concise, I am doing mechanics and, to my surprise, $$\sum_{n=n_0-N+l}^{n_0+N}1 = 2N+1-l$$ Why is it so? Any help clarifying this would be very appreciated.
WorseThanEinstein
  • 187
0
votes
1 answer

Summing something that looks binomial

I have a sum that looks like $\sum_{j=0}^{k/2} \frac{k!}{(k-2j)!j!(2d)^j }p^{k-j}(1-p)^j$. $p \in (0,1), d$ is an integer $> 1$ I am wondering is at least an approximate sum is known.
blanchey
  • 801
1 2 3
…
99 100