Questions tagged [summation]

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

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1answer
89 views

Does $\sum\limits_{k=2}^{\infty}{\frac{|B_{k}|}{k!}(\cos(n)-1)}$ have a closed form?

I am trying to find a closed form expression of the following sum in terms of $n$ (if it exists) where $B_{k}$ is the $k$th Bernoulli number. $$\sum_{k=2}^{\infty}{\frac{|{B_{k}|}}{k!}(\cos(n)-1)}$$ ...
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1answer
44 views

Sum manipulation

my question is if these two sums are equal: $\sum_{j=1}^n \lambda u_jx_j + \mu v_j x_j = \sum_{j=1}^n \lambda u_jx_j + \sum_{j=1}^n \mu v_jx_j$ Thank you for your help.
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3answers
50 views

How to evaluate/calculate sums? [closed]

If you have this expression, $$x + x/2 + x/4 + x/8 + \ldots $$, it equates to roughly $2x$, why is this? Which area of maths is this? To deal with these kind of sums, and getting a result. I am ...
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1answer
36 views

why this statement about $\sum_{i=0}^n a_n$ is false?

Given that ${a_n}$ is positive series, and $\sum_{i=0}^n a_n$ is converge: -There is a sub-series for ${a_n}$ that converge to S>$0$. (THIS statement is false) why is this statement false?
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2answers
59 views

Finite sums in Pascal's triangle [closed]

Today I was solving finite sums in Pascal's triangle see here Here are some examples for finite sums in wikipedia: enter link description here But I couldn't solve these two ones: Prove that $$\...
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2answers
230 views

Derivative of the summation of the log [closed]

Given the function $f(t) = \sum^n_{i=1}{log[(x_i)^{t-1}]}$ where x > 0. What is the derivative of $f(t)$ with respect to t?
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1answer
36 views

Show that $\sum_{k=1}^n e^{2\pi i (k-1)(a-b)/n}=0$

I am trying to show that $$\sum_{k=1}^n e^{2\pi i (k-1)(a-b)/n} = 0$$ where $a$ and $b$ are positive, non-zero integers that run from $1$ to $n$ and $a\neq b$. This is a small part of a larger problem ...
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1answer
212 views

Compute $1^{1/3}+2^{1/3}…7999^{1/3}$ [closed]

How to compute $$1^{1/3}+2^{1/3}.........7999^{1/3}$$
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2answers
112 views

Finding $\sum_{i=1}^{100} a_i$ given that $\sqrt{a_1}+\sqrt{a_2-1}+\sqrt{a_3-2}+\dots+\sqrt{a_n-(n-1)}=\frac12(a_1+a_2+\dots+a_n)=\frac{n(n-3)}4$

Let $a_1,a_2,\dots,a_n$ be real numbers such that $$\sqrt{a_1}+\sqrt{a_2-1}+\sqrt{a_3-2}+\dots+\sqrt{a_n-(n-1)}=\frac12(a_1+a_2+\dots+a_n)=\frac{n(n-3)}4$$ Compute the value of $\sum_{i=1}^{100} ...
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1answer
62 views

Checkerboard Problem

Let $a$ and $b$ be any positive integers, and consider an $a\times b$ checkerboard. Let $S(a,b)$ be the total number of different squares of any size on our $a\times b$ checkerboard. What are the ...
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1answer
67 views

Asymptotic Expansion (Taylor) when n goes to infinity of $ e^{-an}\sum_{k=n+1}^{\infty} \frac{(na)^k}{k!} $

I'm seeking theses asymptotic expansions : $$ e^{-an}\sum_{k=n+1}^{\infty} \frac{(na)^k}{k!} $$ $$ e^{-an}\sum_{k=1}^{n-1} \frac{(na)^k}{k!} $$ in terms of n going to the infinity. Thanks you for ...
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1answer
43 views

Show that the set $A$ contains an open interval.

Let us consider the set $A=\{ \sum_{k=1}^{\infty} \frac{a_k}{5^k}: a_k=0,1,2,3,4\}$$,$ a subset of $\mathbb{R}$. Then show that the set $A$ contains an open interval. My attempt: By the given set $A$...
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1answer
67 views

How to prove $\sum_{k = 1}^{\infty} \sum_{n = 1}^{\infty} n^{- k - 1} (n + 1)^{- k - 1} = \sum_{n = 0}^{\infty} \frac{1}{n (n + 1) - 1}$?

How can we show that $ \sum_{k = 1}^{\infty} \sum_{n = 1}^{\infty} n^{- k - 1} (n + 1)^{- k - 1} = \sum_{n = 0}^{\infty} \frac{1}{n (n + 1) - 1} = 1 + \frac{\sqrt{5}}{5} \pi \tan \left( \frac{\...
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2answers
41 views

solve for x inside of a sum when n is defined (not infinity)

I decided to try this with a simple sum I made $$\sum_{i=1}^{n}{\frac{x}{i}} = 3$$ I know the basic properties of sums, and I tried solving this with the properties; $\sum_{i=1}^{n}{i} = \frac{1}{2}...
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2answers
57 views

How to get this summation in closed form?

$$\sum_{k = 0}^{n} \left( \dfrac{1}{n} - \dfrac{k^{2}}{n}\right)$$ How would one go about putting a summation in closed form if the lower limit is 0 instead of 1? I know you can't immediately use the ...
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1answer
99 views

Showing summations to be equal where the upper limit affects the argument.

NOTE: I do not want the eventual answerer to post a full solution; I only wish that they point me in the right direction as to how to do so. Ok I am trying to prove this for $n>3$: $$2\left(\sum_{...
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4answers
53 views

If n is a positive integer , find the sum of $ S_n = 1+2+\cdots+n$ [duplicate]

$$ S_n = 1+2+\cdots+n=\frac{n(n+1)}{2}.$$ How do I go about solving this ? I tried to search for a solution, but I didn't find anything ( well I didn't know what this was called to even look for it in ...
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1answer
910 views

Probability of the sum of a Poisson Distribution [closed]

Consider a sample $X_1,...,X_{100}$ from a Poisson(3.5) distribution. What is the approximate probability that the total $$T = \sum_{i=1}^{100}X_i$$ exceeds $360$?​ I tried normalizing the function ...
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1answer
501 views

How many integers in the range [10^20, 10^400] exist such that the sum of their digits is a prime number?

I have been trying to solve this problem, but I have absolutely no idea how to do it after an hour of focusing on it. Can someone help me with this? I do not know many summation methods and the ones I ...
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1answer
92 views

The other perspective for the infinite sum of 1+1+1+1+1 [closed]

we use many times using trick for the proof of infinite sum of any sequence or series so, when I was thinking about that then I am confused. Probably it is not formal way to find but I will write in ...
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1answer
84 views

Generating function for sequence $\frac{(-1)^{n+1} B_n(x)}{n}$

I wonder what is generating function for sequence $\frac{(-1)^{n+1} B_n(x)}{n}$. Or, in other words, what is $$\sum_{n=1}^\infty \frac{(-1)^{n+1} B_n(x) t^n}{n}$$ Mathematica fails to provide ...
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1answer
51 views

Finding sum of a series [closed]

How do I find the sum of all fractions whose denominator only have factors of 2 and or 3? Thanks!
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2answers
46 views

Using result for sum of geometric series prove

$\frac{1}{(\alpha+e^{-x^2})}$ = ... using the sum of the geometric series $1 + z + z^{2}+...+z^{n}$ Any help on this would be very appreciated as I'm stuck as to where to start. As in the first ...
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1answer
50 views

If a sum is finite, can you conclude the same for the following sum?

If we know that $\sum_{-\infty}^{\infty} f(x) < \infty$, Then can we conclude that $\sum_{-\infty}^{\infty} x\times f(x) < \infty$? The answer is not immediately obvious to me. I think there ...
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2answers
88 views

How to determine the number of integer solutions to this particular case?

Consider the equation $$z_1 + z_2 + z_3 + z_4 + z_5 + z_6 = k$$ For: $i = 1, \dotsc,6$ $z_i$ is a positive natural number and they must satisfy the following: \begin{align} z_1 & \ge 4 \\ z_2 ...
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2answers
136 views

Maximise population coverage subject to budget constraint

Let $t_i =$ $1$ if transmitter i is to be constructed and $0$ otherwise, $c_j =$ $1$ if community j is covered and $0$ otherwise. Obj func: Max $$z = [10, 15, ..., 10] \cdot c$$ s.t. the ...
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1answer
51 views

What will be the formula of $2^2 + 4^2 + \dots + n^2$? [duplicate]

I'm trying to understand how to calculate $2^2 + 4^2 + \dots + n^2$. I've only succeed to upper bound it by $\dfrac {n^3} 2$. My goal is to say that it is $\Theta (n^3)$. Thank you
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3answers
119 views

How to evaluate the finite sequence (involving the floor function)? [closed]

How to sum this:$$ \lfloor\sqrt{1}\rfloor + \lfloor\sqrt{2}\rfloor + \lfloor\sqrt{3}\rfloor +\lfloor\sqrt{4}\rfloor + \ldots + \lfloor\sqrt{50}\rfloor = ?$$
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1answer
99 views

Expected score of a game [duplicate]

Laertes and Roxane go to the Senate to play a game of Hide-and-Seek. There are 100 rooms in the Senate, and Roxane picks one of them and hides there till the game ends. Laertes, at the beginning ...
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1answer
86 views

What is the value of the series? [closed]

What is the value of the summation $$\dfrac{1}{1!} + \dfrac{1+2}{2!} + \dfrac{1+2+3}{3!} + \dots + {{1+2+3+\dots+i}\over{i!}} + \dots $$ The sum is till infinity.
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1answer
32 views

Summation formulas after altering range.

If i have the following summation to do: $$\sum\limits_{i=1}^{15} (n+1)$$. I know the formula I would use is $$\frac{n(n+1)}{2}$$ and I would sub in $15$ as $n$ and multiply $1$ by $15$. Now if we ...
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1answer
41 views

Show root of unity summation

I have this math problem Let $w$ be a root of unity with $o(w)=n$, with $n > 1$. Show that $$1 + w + w^2 + \cdots + w^{n-1} = 0$$ I'm not entirely sure how to start this problem. Would I ...
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2answers
89 views

How to compute $\sum_{n=1}^N e^{-( n-c)^2}$

I have to compute or at least find good upper and lower bounds on \begin{align*} \sum_{n=1}^N e^{-( n-c)^2/b} \end{align*} and \begin{align*} \sum_{n=1}^N ne^{-( n-c)^2/b} \end{align*} where $c$ ...
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2answers
60 views

Reciprocal over a summation [closed]

Is this statement true? Can we take reciprocal over a summation? $$\frac 1{\sum_{n=1}^\infty\frac 1{(n+1)^3}}=\sum_{n=1}^\infty (n+1)^3$$
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1answer
21 views

Is it possible to find $n$ from the sum forming a polynomial?

How does one solve for $n$ in: $100000 = \sum\limits_{x=1}^n 1020.2065\ x^{-0.3431}$
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4answers
109 views

Deriving formula for sum n(n+1) [duplicate]

Can you please describe how to derive a formula for first n members of $$ S = 1\cdot 2 + 2\cdot 3 + 3\cdot 4 +\cdots +n(n+1)\mbox{?} $$ Thank you
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1answer
303 views

Generalized Holder Inequality

Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ ...
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1answer
99 views

Does grouping the terms of a series (but not moving them) change the sum?

This question is an extension of this one, in which I am told that, given a sequence $a_1, a_2, a_3, ...$, $$\sum_{j=1}^{\infty }a_{j}=\sum_{n=1}^{\infty }(\sum_{k=2^{n-1}}^{2^{n}-1}a_{k})$$ is only ...
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2answers
51 views

Is the series $\sum \frac{3 + \sin n}{n^2}$ convergent?

How can I show if the following series converges? $$\sum \frac{3 + \sin n}{n^2}$$ I can't use differential or integral calculus (hasn't been covered in my class yet.)
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2answers
58 views

How to find the value of $\sum_{k=1}^\infty (\frac{1}{9})^k$ using partial sums?

So I was trying to prove an infinite sum by looking at the partial sum, when I ran into a problem. Consider: $$\sum_{k=1}^n \left(\frac{1}{9}\right)^k = \frac{1}{8} 9^{-n} (9^n-1)$$ but as there are ...
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1answer
188 views

How to prove that Grandi's series $= \frac{1}{2}$ using Euler transform [closed]

Let $x$ denote Grandi's series $1-1+1-1+1-1+1-...$ This implies that $$ x = 1\text{ or}\\ x = 0\text{ or}\\ 1-x = 1 - (1-1+1-1+1-...) = x \implies 2x = 1 \implies x = \frac{1}{2}$$ Where the last ...
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2answers
575 views

Recurrence Relation Solving Problem

Can anyone help me in solving this complex recurrence in detail? $T(n)=n + \sum\limits_{k-1}^n [T(n-k)+T(k)] $ $T(1) = 1$. We want to calculate order of T. I'm confused by using recursion tree ...
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1answer
42 views

Is there an expression for $(S/k)$ where $S=\sum_{n=1}^\infty n$ and $k \in \mathbb{Z}$?

Given that $S=\sum_{n=1}^\infty n=-1/12$ (for an explanation see this question or this video from Youtube) For example if $k=4$: $(S/4)=1/4+2/4+3/4+1+5/4+6/4+7/4+2+9/4...$ Please edit to improve ...
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2answers
31 views

Compute the sum $\sum_{i=0}^n 5^{i+1}-5^i$

Compute the sum: $$\sum_{i=0}^n 5^{i+1}-5^i$$ with the hint, "start by writing out (and expanding) the sum." So I did and got $$4 + 20 + 100...$$ with the appearance of going to infinity. Is ...
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votes
3answers
379 views

Prove that $2\sum_{k=1}^n\cos kθ= \frac{\sin\left(n+\frac12\right)θ}{\sin\fracθ2}-1$ [closed]

Prove that $$2\sum_{k=1}^n\cos kθ= \frac{\sin\left(n+\frac12\right)θ}{\sin\fracθ2}-1$$ By using $$e^{iθ}+e^{2iθ}+\cdots+e^{niθ}=\frac{e^{iθ}(1-e^{inθ})}{1-e^{iθ}}$$
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1answer
665 views

Find the biggest sum from sequence of number which within a range

I need help. How do I find the greatest sum from sequence of number within a finite range, for example: Given sequence {2,5,4,3,6} and the range is 11, so how to find the number within the sequence ...
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votes
1answer
126 views

Proof of $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$ [duplicate]

Prove that $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$ I have actually got the HINT for this Proof from a very nice book: The HINT i started with is: $$\frac{1}{\sin^2 x}=\frac{1}{4\sin^2\...
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1answer
29 views

Differential Geometry - vector fields Lie bracket

Can anyone tell me why (in the last line) $i$ changes to $j$ in the first component of the sum?
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1answer
44 views

Summing a series exactly 1 [duplicate]

How does one go about exactly summing this series $$\sum_{n=1}^{\infty}\frac{(-1)^nn^2}{3^n}$$ Stuck on this and not sure how to proceed. Appreciate any assistance!
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2answers
34 views

The taylor series is given determine $a_n$ for which x converges to f(x)

The taylor series of the function f(x) = $1-\mathrm{e}^{-x^2}$ around x = 0 is given by $\sum_{n=0}^{\infty} a_nx^n$ determine $a_n$ for all n $\geq$ 0 and give for which value of x the series ...