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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Rewriting summation formula

I just watched a tutorial on recurrence by substitution. In the tutorial, it mentioned about rewriting $\sum\limits_{i=1}^\mathbb{k}{2^i}$ as (2k+1 - 2). My question is can I generalize it as xlimit + 1 - x where x is the base.
Andrew
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What is the integer part of $\sum_{i=2}^{9999} \frac {1}{\sqrt i}?$

What is the integer part of: $$\sum_{i=2}^{9999} \frac {1}{\sqrt {i}}$$ A short but tricky problem. Any help is welcome. Source
Soham
  • 9,990
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Double summation index meaning

I am trying to find a book or something that explains the use of a inequality in the index notation of a double summation like in this example: $$\mathop{\sum\sum}_{i\leq j}Cov(X_i,X_j).$$ What does this mean? Does not seem intuitive for me a like a…
Carlos Lobato
  • 139
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Subset Sum Problem Variation?

There are $100$ cards with a unique number from $1$ to $100$ written over them. How many ways can someone pick exactly $5$ cards where the numbers on them sum to $100$? I am not sure but this could be a variation of the subset sum problem. But one…
Shaown S.
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Find the sum : $\displaystyle\sum_{i=0}^n \frac{2^i}{1+x^{2^{i}}}$

$\displaystyle\sum_{i=0}^n \frac{2^i}{1+x^{2^{i}}}$ What technique is applicable here? I can't find a way to manipulate this sum to make it telescope. Just guide me.
Adienl
  • 1,069
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Sum of $p\sum_{i = 1}^{\infty} i(1 - p)^{i - 1}$

Does anyone know how to find the exact sum of $$ p\sum_{i = 1}^{\infty} i(1 - p)^{i - 1} $$
Geoff Smith
  • 191
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How find this sum $\sum_{n=1}^{\infty}n\sum_{k=2^{n-1}}^{2^n-1}\frac{1}{k(2k+1)(2k+2)}$

Find the sum $$\sum_{n=1}^{\infty}n\sum_{k=2^{n - 1}}^{2^{n}\ -\ 1}\dfrac{1}{k(2k+1)(2k+2)}$$ My try: note $$\dfrac{1}{k(2k+1)(2k+2)}=\dfrac{2}{(2k)(2k+1)(2k+2)}=\left(\dfrac{1}{(2k)(2k+1)}-\dfrac{1}{(2k+1)(2k+2)}\right)$$ Then I can't
math110
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How to prove that $\sum_{n=1}^{231}\sec^{4}\left(\frac{2\pi n}{231}\right)=949167681$?

Prove the following: $$\large\sum_{n=1}^{231}\sec^{4}\left(\frac{2\pi n}{231}\right)=949167681$$ Trying to add the sum to itself doesn't do much and I cannot telescope it easily, and I got lost using trigonometric identities but the $\frac{2\pi}{N}$…
FreshestCheese
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Finite Power Sum Derivation

TL;DR: "I've made something work but I do not know why it works - please explain it to me (or disprove my finding)" DISCLAIMER: A lot of things here are done in a non-rigorous way - which is all the more surprising that in the end it "all…
Alma Do
  • 499
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Pulling constants out of a summation

Is there anyway I can pull x out of this summation to solve for it? $$ 0 = \frac{-2}{N} \sum_i^N a_i(b_i - (xa_i))$$ EDIT: Thanks for all the quick comments guys. I am confused by a certain step everyone seems to be taking. $$ -\sum_i^N xa_i^2 =…
still learning
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Series with $k^2$ coefficients

Let $\{S_k\}$ be a sequence of numbers. Then by reversing the sum we have the following, $$A=\sum_{k=0}^n k\cdot S_k \cdot S_{n-k} = \sum_{k=0}^n (n-k) \cdot S_{n-k} \cdot S_{k} .$$ Thus adding these sums together gives us, $$A+A=\sum_{k=0}^n n\cdot…
Bobby Ocean
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Formal Definition of Summation over a Set

Formal definition of summation over a sequence is very clear. Suppose we have a sequence $\left\{a_{i}\right\}_{0}^{\infty}$. Then the summation of this sequence can be defined as a new sequence $\left\{ b_i \right\}_0^\infty$ in a recursive…
Ziqi Fan
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How to make sense of a weird Summation?

I have been toying with some sums, as I saw that $123456789\div987654312 = \frac{1}{8}$. Which led me to wonder if this can be generalized to other bases. I.E. in base 4 it would be…
Alex
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How to find that limit by mathematica?

Limit[Sum[2(2k)^(1/(2k))-k^(1/k),{k,n+1,2n}]-n, n -> ∞] to solve by hand, $$\sqrt[y]y=e^{\frac{\ln y}{y}}\sim1+\frac{\ln y}{y}$$ $$2\sqrt[2k]{2k}-\sqrt[k]k\sim1+\frac{\ln2}{k}$$ $$\sum_{k=n+1}^{2n}\left(2\sqrt[2k]{2k}-\sqrt[k]k\right)\sim…
y-ung'1 0'm-o
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Find the value of the expression.

$$ \sum_{i=1}^{1000} i^\frac{-2}{3} =M$$ $$\text{Then find the value of [M]-20} $$ $$\text{where[ . ] Denotes the greatest integer function.}$$ My attempt, I was able to figure out the lower end of M as $$\sum_{i=1}^{1000} i^\frac{-2}{3} \gt…
DareDevil
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