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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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2answers
66 views

What is the limit of $\sum_{k=1}^{n}\frac{k^3}{n^4}$? [closed]

Find the following limit: $$\lim_{n \to \infty} \sum_{k=1}^{n}\frac{k^3}{n^4}$$ Should I approximate this with integrals?
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votes
1answer
76 views

$\sum\limits_{n=1}^{\infty}\sin ( \frac{n}{2^n})$ converges? [closed]

I was trying to determine weather or not $\sum\limits_{n=1}^{\infty}\sin ( \frac{n}{2^n})$ converges using perhaps the D'Alembert test, but given the sine I cant really see it happening..are there ...
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votes
3answers
108 views

Induction proof of $1+3+\cdots+3^n=\frac{3^{n+1}-1}{2}$ [closed]

How would I prove the following by induction?$$1+3+3^2+3^3+\cdots+3^n=\frac{3^{n+1}-1}{2}$$ for all $n\geq 0.$ I kept trying to create a base case but I am not sure how many I need. I also seem to be ...
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votes
2answers
72 views

How can we prove that $27$ is the largest number $χ$ such that $\sum_{i \in δ(χ^3)} = χ$? [closed]

According to Stetson University, $27$ is the largest number $χ$ such that $\sum_{i \in δ(χ^3)} = χ$ where $δ(χ) =$ digits of $χ$. I have tried proving that this is true, but do not know where to ...
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votes
1answer
246 views

How do I convert $1 - 1 + 1 - 1 + …$ to summation notation?

I can convert $1 + 2 + 3 + 4 + 5 + ... = -\frac {1}{12}$ to summation notation: $$\sum_{n = 1}^\infty n = -\frac {1}{12}$$ But, how can I convert the following series to summation notation: $$1 - 1 + ...
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votes
1answer
93 views

Value of sum of binomials: $P = \binom{N}{0}-\binom{N}{1}+\binom{N}{2}-\binom{N}{3}+ \dotsb + (-1)^N\binom{N}{N}$ [duplicate]

$P = \binom{N}{0}-\binom{N}{1}+\binom{N}{2}-\binom{N}{3}+ \dotsb + (-1)^N\binom{N}{N}$ I can calculate the value of this equation manually, but there any direct formula for calculating the value of ...
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votes
1answer
48 views

Solving mathematical summation [closed]

How to solve the following sum: $$\sum_{i = 0}^n \frac{(1/2^n)^i \cdot (1 - 1/2^n)^{n-i}}{i!\cdot(n-i)!}$$
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votes
3answers
67 views

Find the sum of all products of two distinct naturals, neither exceeding 2015. [closed]

Find the sum $$(1\cdot2)+(1\cdot3)+(1\cdot4)+\cdots+(1\cdot2015)+(2\cdot3)+(2\cdot4)+\cdots+(2\cdot2015)+\cdots+(2014\cdot2015)$$ any help? I tried with telescope but got nothing
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votes
2answers
59 views

Is the sum of the following series a finite number or not? Explain. $ \sum_{k=1}^\infty \frac{5\sin^2k}{k!} $ [closed]

Is the sum of the following series a finite number or not? Explain. $$ \sum_{k=1}^\infty \frac{5\sin^2k}{k!} $$
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votes
1answer
54 views

Sum of finite series given sum of cubes [closed]

The question says - If $1^3+2^3+3^3+\cdots+10^3=3025$, then what is the value of the following series which is ? $$4+32+108+\cdots+4000$$
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votes
3answers
73 views

$\lim_{n\to \infty}\dfrac{\sum\limits_{k=2}^n k\cos\dfrac{π}{k}}{n^2} $ [closed]

Find the value of $$\lim_{n\to \infty}\dfrac{\sum\limits_{k=2}^n k\cos\dfrac{π}{k}}{n^2} $$ Please help me, I can't find any clues..
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votes
2answers
73 views

How one can simplify this product of sums

My question is simple: How one can simplify this product of sums: $$S=\left(\sum_{k=1}^{n}a_{k}\right) \left(\sum_{k=1}^{p}b_{k}\right) \left(\sum_{k=1}^{q}c_{k}\right)$$ where $a_{k},b_{k},...
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votes
3answers
84 views

How is this series rearranged?

I'm stuck at this. How is RHS rearranged? Is it a change of index? $$ \sum_{n=1}^{2N} \frac{1}{n} - \sum_{n=1}^{N} \frac{1}{n} = \sum_{n=N+1}^{2N} \frac{1}{n} $$ I'm stuck here: $$ \sum_{n=1}^{2N} \...
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votes
2answers
71 views

Sum of numerators divided by sum of denominators $\leq$ the maximum fraction [duplicate]

Let $\tfrac{a_1}{b_1},\dots,\tfrac{a_n}{b_n}$ where $a_i,b_i>0$. How can one prove that $$\frac{\sum_i a_i}{\sum_i b_i}\leq \max_j \tfrac{a_j}{b_j}$$?
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votes
2answers
58 views

$\sum_{n=1}^\infty x(1-x)^{n-1}$ Does this sum converge uniformly?

$$\sum_{n=1}^\infty x(1-x)^{n-1}$$ I know that this sum converge $\iff$ $0\le x \le 1$, i wanted to use the Weierstrass but could not suceed, so i think this sum might not converge uniformly,but i'm ...
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votes
1answer
64 views

Summing this series??

Let $T_n$ = $\sqrt{2000^2 - n^2}$ The sum of this series to n terms is $S_n$ Then find $\frac{2}{2000^2}(2000 + 2S_{2000})$ upto $2$ decimal places. I know in homework questions we do have to ...
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1answer
108 views

Remainder of $\sum_{x=1}^{312} x \times x!$ divided by 2016 [closed]

I have this question: What is the remainder of $$\sum_{x=1}^{312} x \times x!$$ (or just simply) $$(1! \times 1) + (2! \times 2) + (3! \times 3) + \dots + (312! \times312)$$ Divided by ...
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votes
2answers
65 views

A Riemann-type sum [closed]

I want to solve this summation, however I have no idea where to start. Could any one help me find a good starting place? $$ \sum_{i=1}^{n}\sin\left(i \over n\right)\frac{1}{n} $$ Any help would be ...
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votes
1answer
354 views

$\sum_{i=1}^n \frac{n}{\text{gcd}(i,n)}.$ [closed]

Find the value of this series: $$\sum_{i=1}^n \frac{n}{\text{gcd}(i,n)}.$$
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votes
2answers
832 views

Show that $\sum \limits_{i=1}^n log(i)$ is O(n log n) [duplicate]

How can I show that $\sum \limits_{i=1}^n \log(i)$ is $O(n \log n)?$ (Log in base 2).
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votes
3answers
804 views

How to simplify this triple summation

I am trying to calculate the following summation by n : $$\begin{align} \sum_{i=1}^n \sum_{j=i}^n \sum_{k=i}^j 1 &= \sum_{i=1}^n \sum_{j=i}^n (j-i+1) \\ &= \sum_{i=1}^n \left( \sum_{j=i}^...
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votes
1answer
33 views

$n k^n$ summation question [closed]

How does one prove that Can this be extended to higher powers such as: Thanks!
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votes
1answer
797 views

How to derive this summation formula? [closed]

$$\sum_{i=0}^{n-1} ia^i = \frac{a-na^n+(n-1)a^{n+1}}{(1-a)^2}$$ What is the thought process behind obtaining this formula?
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votes
1answer
71 views

Find the Value of $\sin^2A + \sin^2B + \sin^2C$ given the following data. [duplicate]

If $2\tan^2A\tan^2B\tan^2C + \tan^2A\tan^2B + \tan^2B\tan^2C + \tan^2C\tan^2A = 1$, then find the value of $\sin^2A + \sin^2B + \sin^2C$. My attempt 1). I tried to multiply both sides by $\cos^2A\...
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votes
2answers
35 views

Hey want a detailed hint about how to solve this question [closed]

The question is: To prove the following summation for every positive integer n: $$\sum_{k=0}^n x^k = ({n+1}){..if..x =1 }$$ and a similar one which is: $$\sum_{k=0}^n x^k = \frac{x^{n+1}-1}{x-1} {.....
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votes
1answer
34 views

What is the sum $\sum_{k=10}^{\infty}\left(\frac{1}{2x}\right)^k$ [closed]

I would appreciate some directions regarding the follow problem, $\sum_{k=10}^{\infty}\left(\frac{1}{2x}\right)^k=$?
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votes
1answer
49 views

prove the statement [closed]

Statement:If $a_1,a_2,a_3\cdots a_n$ be $n$ unequal and positive quantities and if $m>r>0$ , then $$\frac{a_1^{m}+a_2^{m}\cdots +a_n^{m}}{n}> \frac{a_1^{r}+a_2^{r}\cdots +a_n^{r}}{n}. \frac{...
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votes
1answer
68 views

Question concerning sigma notation [duplicate]

Consider you have been given that $$\sum_{i = 1}^{\infty}i = -\dfrac{1}{12} $$ How do you solve this sigma notation? I've not seen this kinda sigma notation before. Regards!
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votes
1answer
53 views

Please simplify this sigma question? [closed]

I am not able to solve this sigma question. Please anybody solve this question by steps. $$\sum_{R=1}^N\left(\frac13\right)^{R-1}$$
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votes
3answers
407 views

Series into sigma notation

How do I convert $-1 + 3 - 5 +...- 101$ into sigma notation. I tried to divide the series into $-1 -5 -7 -...$ and $3+7+9$ but i'm not too sure if that is correct.
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votes
2answers
51 views

Evaluating the following sums with these suppositions. I really need help with this. [closed]

I have tried everything and I am just unable to solve the following sums. Mainly because I do not understand the suppositions and why they are there in the first place and further I do not get what ...
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votes
2answers
57 views

Find the partial sum of $S_n$ [closed]

EDİT: How we can calculate $S_n$ for any $k$ with MATRİX METHOD? $S_n=1^k+2^k+3^k+...+n^k$ $k\in$$\mathbb{N}$
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votes
1answer
38 views

Solve $\sum_{i=1}^{200} {1\over{1+x_i}} =?$ [closed]

$$ (x^{2}+x+1)^{100}=a_0+a_1x+a_2x^{2}+...+a_{199}x^{199}+a_{200}x^{200}$$ $$\sum_{i=1}^{200} {1\over{1+x_i}} =?$$ Can somebody help me? Thank you!
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votes
1answer
832 views

what is the summation from i=0 to log(n) [closed]

I need to know how to get the summation of a constant (c) from i=0 to log(n) of a constant
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votes
1answer
186 views

$((n-1)^{0.5})/(((n+1)^2)-1)$ Is the sum convergent?, why or why not? [closed]

$$\frac{(n-1)^{0.5}}{(n+1)^2-1}$$ Sorry I dont know how to to do sub or superscripts. I would like a step by step method please, thanks.
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votes
2answers
46 views

Prove by induction that $\sum_{i=1}^n i \geq \frac{n^2}{2}$ [closed]

Can someone show me a formal proof of this exercise ? \begin{equation} \sum\limits_{i=1}^n i \geq \frac{n^2}{2}, \quad \forall n \in \mathbb{N}. \end{equation} Thanks to anyone who can help! :)
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1answer
2k views

Sum of finite series involving square roots [duplicate]

What's the the result of: $$\sum_{k=1}^{n}{\sqrt{k}+1}$$ Thanks.
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2answers
75 views

Summation Problems [closed]

How did this particular equation come about? I haven't seen it before in the summation rules index on wikipedia: $$\sum\limits_{i=1}^{k+1} x_i =\left(\sum\limits_{i=1}^{k} x_i\right)+x_{k+1} $$ Edit:...
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votes
2answers
250 views

How to use math symbols to represent a basic formula

I've a fomula that looks like this: How to properly represent this formula using SUM Symbol and MEAN symbol? $$ r=2 \left(3(f_1) +\frac{(g_1+...+ g_n)}{n} + \frac{s_1 +... + s_n}{n}\right) + x_1 +...
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votes
1answer
790 views

Sigma of factorial function? [duplicate]

How would you find this sum mathmatically?$$\sum_{i=0}^\infty \left( \dfrac{1}{2} \right) ^{i!}$$ What techniques would you use to solve this as well?
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votes
1answer
55 views

How to convert an infinite summation to an integral [closed]

I need help converting this summation to an integral $$\sum_{i=1}^\infty 1+\frac{\vert(n-im)\vert}{(n-im)}$$ I keep trying but get stuck so any help is appreciated.
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votes
1answer
38 views

Upper and lower bounds for series and sequences [closed]

Find upper and lower bounds for the following finite sum: $$1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}}$$
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votes
1answer
32 views

Index summation over a binomial coefficiant series [closed]

Can the following sum be reduced to an analytic expression? $$\sum_{m=0}^N\frac{N!}{m!(N-m)!}$$
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votes
1answer
31 views

$\sum_{i=0}^{2n-1} e^\frac{i}{n}$?

I was doing a Riemann integral of $\int_0^2e^xdx$ and ended up with $\overline S(\mathcal{D})= \frac{e^{\frac{1}{n}}}{n} \sum_{i=0}^{2n-1}e^\frac{i}{n}$. I was wondering how you would compute this ...
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votes
3answers
48 views

Find a closed expression for $\sum_{k=1}^n\left(\sum_{l=1}^k l^2\right)$. [closed]

Find the close expression for : $$\sum_{k=1}^n\left(\sum_{l=1}^k l^2\right)$$ I have very less knowledge about nested sigma notations. Any help or hints would be greatly appreciated.
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votes
1answer
103 views

Evaluating the sum of $\sum_{k=0}^n \binom{n}{k}^2$ [duplicate]

I don't know if this question is trivial but let me put it in the first place. I'm trying to find the sum of $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n-1}^2+\binom{n}{n}^2$ or ...
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votes
2answers
302 views

Finite sum k*x^k [closed]

Prove that $$\sum_{k=0}^m k x^k = \frac{x(mx^{m+1}-(m+1)x^m +1)}{(x-1)^2}$$.
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votes
1answer
27 views

Summation simplification in the definition of composition of linear transformation

I'm reading a definition of composition of linear transformation, and I don't understand the second line in it. And I think that the part I don't understand is not directly related to linear ...
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votes
1answer
203 views

simplification of double summation [closed]

I have solved double summation problem,.Kindly check it whether it is correct or not?? $$\sum_{j=1}^3\sum_{i=1}^j (i+j) = 12$$ thanks
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votes
1answer
42 views

How to convert this sum into an equation? [closed]

How to convert this sum into an equation? $$ \sum_{i=1}^{C-1} i(R-i)(C-i) $$ Expanded: $$ \sum_{i=1}^{C-1} (RCi - Ri^2 -Ci^2 + i^3) $$ Which can be represented as: $$ RC\sum_{i=1}^{c-1}i - R\...