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

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1
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3answers
43 views

How do I find a solution to this this finite series? $ \frac{1}{n^4} \sum_{i=1}^{n} \left({i^3}\right) $

How do I find a solution to this this finite series? Any help would be greatly appreciated. $$ \frac{1}{n^4} \sum_{i=1}^{n} \left({i^3}\right) $$
-1
votes
0answers
48 views

How to simplify $\sum_{r=1}^{y} \binom{x-1}{r}\binom{y-1}{r}$? [on hold]

To find sum of the product of two combination terms $$\sum_{r=1}^{y-1} \binom{x-1}{r}\binom{y-1}{r}$$
0
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1answer
29 views

to simplify the following combinatorial terms [on hold]

To simplify the following summation involving product of combinations $$\sum_{r=1}^{y}\left(\begin{array}{c} x-1 \\ r \end{array}\right) \left(\begin{array}{c} y-1 \\ r-1 ...
-4
votes
2answers
51 views

Proof By Induction [on hold]

I am trying to prove the Following, However, I dont understand what to do at the Inductive Step: Any Help would be appreciated!
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0answers
22 views

Find Functions That Can Be Inverted from Their Sums

I have the following situation:$$ f_1(x_1) + f_1(x_2) + f_1(x_3) + \cdots + f_1(x_n) = c_1\\ f_2(x_1) + f_2(x_2) + f_2(x_3) + \cdots + f_2(x_n) = c_2\\ \vdots\\ f_n(x_1) + f_n(x_2) + f_n(x_3) + ...
1
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1answer
31 views

Proving the equality of sequences' limit [duplicate]

Prove that if $a_n$ is convergent, then $M_n:=\frac{1}{n}\sum_{1}^{n} a_n$ satisfies $\lim M_n=\lim a_n$. (sorry for English)
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1answer
25 views

Summation. Combining different set of indices.

I am reading the second chapter of Concrete Mathematics book and I cant get my head aroud a simple concept: it is stated there that $$ \sum _{k \in K} a_k + \sum _{k \in K'}a_k = \sum _{k \in K \cap ...
2
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1answer
104 views

How can I prove this combinatorial identity?

Let $n,m$ be non-negative integers. How can one prove the following identity? $$\sum_{j=0}^n j\binom{2n}{n+j}\binom{m+j-1}{2m-1}=m\cdot4^{n-m}\cdot\binom{n}{m}$$
1
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0answers
10 views

Dealing with constants in summation notation

I haven't worked with summation notation in a while, and am unsure how to approach the following: $\sum_1^n [-\frac 12 * \frac{(x_i - \alpha)^2}{\alpha}]$ where $\alpha \in R$ What would be the ...
0
votes
2answers
32 views

Calculating a power series [on hold]

I was wondering if anyone knows how to calculate: $\sum_\limits{t=-\infty}^{\infty}$ $a^{t} e^{-itb}$, for constants a,b and $-\pi < b < \pi$ Can we take the t=0 term out to reduce it ...
0
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0answers
43 views

Infinity Series [on hold]

Good afternoon. I'm brazilian, then sorry by my bad english. I have a problem with one question about Infinite Series. I searched for anyone method could help me. I have all constants values (w, y, ...
1
vote
0answers
24 views

Infinite series

Good afternoon. I'm brazilian, then sorry by my bad english. I have a problem with one question about Infinite Series. I searched for anyone method could help me. I have all constants values (w, y, ...
2
votes
0answers
29 views

Sum of Floor of Square Root: $S = \sum_{k=1}^{n} \lfloor \sqrt{k}\rfloor$

$$ S = \sum_{k=1}^{n} \lfloor \sqrt{k}\rfloor. $$ Hello, I´m trying to solve this summation. I was able to get $$ a_n = 2a_{(n-1)} - a_{(n-2)} $$ for non perfect square numbers and $$ a_n = ...
0
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0answers
21 views

inclusion–exclusion principle formula in probability for general case

For the system reliability analysis of complex systems, I'll use the formula below after a point in a custom computer code. I know that for the n=2 case, formula result is $P_1+P_2-P_1.P_2$ ($P$ : ...
-1
votes
3answers
39 views

Summation of $3^k$ from $2$ to $72$ [on hold]

i'm currently stuck on the following question and am not supposed to be using a summation calculator to find the answer: $$ \sum _{k=2}^{72}\left(3^k\right) $$ Please could somebody explain to me ...
1
vote
1answer
19 views

Sum identity involving sin

How one can prove that $$\sum_{k=1}^n(-1)^k\sin(2k\theta)=\cos(n\pi/2+\theta+n\theta)\sec\theta\sin(n\pi/2+n\theta)?$$ It looks difficult as there is sum on the other side and product of trigonometric ...
1
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0answers
42 views

Summation of trigonometric series

The first part of the question requires me to show that the sum of $$cos(2n - 1)x = \frac{sin(2Nx)}{2sin(x)}$$ from $n = 1$ to $N$. This I have done by considering the real part of the geometric ...
0
votes
3answers
81 views

approximation of a sum $\sum ^{\sqrt{n} }_{k=5}\frac{\log\log(k)}{k\log(k)} $ [on hold]

What is the approximation of this sum $$\sum^{\sqrt{n}}_{k=5}\frac{\log\log(k)}{k\log(k)}$$ Should I proceed by an integral ? How can I calculate its lower and upper bound?
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0answers
29 views

Asymptotic Formula for Sum

I am trying to find an asymptotic formula for the sum of the following: $\sum _{x=1}^{\infty } x \left(\left(1-\frac{\Gamma (x,\lambda )}{\Gamma (x)}\right)^n-\left(1-\frac{\Gamma (x+1,\lambda ...
0
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0answers
21 views

Combinatorial identity binomial coefficients [duplicate]

How to prove that $$ \binom{m}{p} = \sum_{j=0}^q \binom{q}{j}\binom{m-q}{p-j}\;?$$
3
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2answers
19 views

notation for Sumation of Sumation for only for odd iterations

I need to write a summation in summation whether the inner summation should iterate from one through all odd numbers to the teration of the outer summation which goes from 1 to $\infty$... Something ...
0
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0answers
22 views

Symbol for exponentiation of a sequence? (Equivalent to SIGMA for summation and PI for product)

I have a student asking whether there is a symbol for exponentiation of a sequence? So there's SIGMA for summation of a sequence, PI for multiplication of a sequence and perhaps something else for ...
2
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1answer
40 views

An expression with gcd and abs is transformed magically!

There's a problem to calculate $\sum^{n}_{i=1}\sum^{m}_{j=1}\frac{|i-j|}{\gcd(i,j)}$, whose tutorial gives the following transformation I really don't understand. ...
3
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4answers
72 views

How to prove $ \sum\limits_{k=1}^{n}\frac{k}{(k+1)!}=1-\frac{1}{(n+1)!}$ using induction?

This is as far as I get. I get stuck here because both sides to not equal each other, but I am not sure what I am doing wrong. $$ \sum\limits_{k=1}^{n}\frac{k}{(k+1)!}=1-\frac{1}{(n+1)!}$$ Assume: ...
2
votes
3answers
55 views

Show the closed form of the sum $\sum_{i=0}^{n-1} i x^i$ [duplicate]

Can anybody help me to show that when $x\neq 1$ $$\large \sum_{i=0}^{n-1} i\, x^i = \frac{1-n\, x^{n-1}+(n-1)\,x^n}{(1-x)^2}$$
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1answer
35 views

Need help using ratio test [on hold]

Only using the ratio test determine where the series converges. $$\sum_{n=1}^\infty \frac{8n!}{n^n}$$
2
votes
1answer
37 views

Find closed form formula

I need help to find closed form formula for this summation $$\sum_{i=0}^{\infty}(x-y)^i$$
6
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6answers
952 views

If each term in a sum converges, does the infinite sum converge too?

Let $S(x) = \sum_{n=1}^\infty s_n(x)$ where the real valued terms satisfy $s_n(x) \to s_n$ as $x \to \infty$ for each $n$. Suppose that $S=\sum_{n=1}^\infty s_n< \infty$. Does it follow that ...
3
votes
1answer
40 views

Help understanding the complexity of my algorithm (summation)

As an exercise, I wrote an algorithm for summing the all elements in an array that are less than i. Given input array A, it produces output array B, whereby B[i] = sum of all elements in A that are ...
1
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1answer
35 views

It is okay to have a conditions in a summation limit that depend on the current value of another summation

this is one of those things that I know how to do in a programming environment but not sure how it translates into mathematics. I'm trying to express a sum so that it is easily visible that certain ...
1
vote
2answers
70 views

How to expand $ \sum\limits_{k=1}^{5}\frac{k}{(k+1)!}$ [on hold]

Can somebody explain step by step how to expand $$ \sum\limits_{n=1}^{5}\frac{k}{(k+1)!}$$ into an equation?
1
vote
2answers
47 views

this sum inequality $\sum_{i=1}^{n}\frac{1}{4i(i+1)-1}<\frac{2}{7}$

show that $$\sum_{i=1}^{n}\dfrac{1}{4i(i+1)-1}<\dfrac{2}{7}\tag{1}$$ we have $$4i(i+1)-1>4i^2$$ But $$\sum_{i=1}^{n}\dfrac{1}{i^2}<\dfrac{8}{7}$$ it is clear not hold, because ...
0
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0answers
23 views

Simplifying a formula with summation

I have seen on a data-mining tutorial this sort of formula simplification: to (consider $\bar x$ to be the sum of all $x_i$ dividied by n) So I was wondering if I can apply the same ...
0
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1answer
47 views

Closed form for the summation [on hold]

Can anyone help me to find what is closed form formula for this summation formula $$\sum\limits_{n=0}^mx^n\sum\limits_{i=0}^ny^i$$
1
vote
1answer
53 views

How many unique ways can I sum $k$ non-negative numbers to $N$?

I have a similar question but not exactly the same as this. I'm trying to determine the number of unique multisets $S\in \mathbb{N}$ that exist when the members are required to sum to a number $N$. ...
1
vote
1answer
30 views

If $ f(n) = \sum_{i = 1}^{n} (n / i) \log(n / i) $ and $ g(n) = n ~ {\log^{2}}(n) $, then is $ O(f) = O(g) $?

I was trying to prove that if $$f(n) = \sum_{i=1}^{n}\frac{n}{i} \log\frac{n}{i} $$ $$g(n) = n \log^2n$$ then $O(f(n)) = O(g(n))$ I am not sure that it is the case, but based on my simulation ...
1
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0answers
39 views

2015 AMC12A question 25

This is a question from the 2015 AMC12 math competition. I haven't really made much progress at all on it, and I just want to know the right way to solve this equation. A collection of circles in ...
0
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2answers
26 views

solving the limit $\lim_{n\to \infty}\sum_{k=1}^n|e^{(2πik)/n}-e^{(2πi(k-1))/n}$|

$$\lim_{n\to \infty}\sum_{k=1}^n\left|e^{(2πik)/n}-e^{(2πi(k-1))/n}\right|$$ i can solve it geometrically. but is there any way to solve it using Euler's formula ?, the answer will be one of these ...
3
votes
3answers
72 views

Does $\sum_{n=1}^\infty \frac{\cos(n\pi/3)}{n!}$ absolutely converge?

Using the Ratio Test, I have to find whether $$ \sum_{n=1}^\infty \frac{\cos(n\pi/3)}{n!} $$ converges or diverges. The back of the book says that the sum is absolutely convergent. My work: $a_n = ...
3
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2answers
66 views

Show that if $a\neq 1$, then $\sum_{k=0}^{n-1}ka^k = \frac{1-na^{n-1}+(n-1)a^n}{(1-a)^2}$

Need to show that if $a\neq 1$, then $$\sum_{k=0}^{n-1}ka^k = \frac{1-na^{n-1}+(n-1)a^n}{(1-a)^2}$$ Here is my attempt: $$\begin{aligned} S & =\sum_{k=0}^{n-1}ka^k \\ &= ...
3
votes
2answers
56 views

Find the closed-form for $\sum_{i=0}^n(-1)^i(\frac{1}{2})^i$

I start with simplifying: $$\sum_{i=0}^n(-1)^i(\frac{1}{2})^i=\sum_{i=0}^n(-\frac{1}{2})^i$$ then: $$S = 1 + (-\frac{1}{2}) + (-\frac{1}{2})^2 + ... +(-\frac{1}{2})^n$$ $$(-\frac{1}{2})S = ...
1
vote
1answer
15 views

What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function: $$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$
4
votes
4answers
93 views

Prove $\sum_{i=2}^{n}\frac{1}{(n-1)n}$ = $\frac{(n-1)}{n}$ using induction.

I need to prove $\sum_{i=2}^{n}\frac{1}{(i-1)i}$ = $\frac{(n-1)}{n}$ using induction. I am getting stuck midway through the inductive step. Here is what I have: $\forall n\geq 2$, where ...
3
votes
4answers
78 views

Sum of Square roots formula.

I would like to know if there is formula to calculate sum of series of square roots $\sqrt{1} + \sqrt{2}+\dotsb+ \sqrt{n}$ like the one for the series $1 + 2 +\ldots+ n = \frac{n(n+1)}{2}$. Thanks in ...
0
votes
4answers
53 views

Solution verification: $\sum_{n=1}^\infty \frac{9^n}{3+10^n}$

I need to find out whether $$\sum_{n=1}^\infty \frac{9^n}{3+10^n}$$ converges or diverges using the limit comparison test. Here's my work: Let $a_n$ be $\frac{9^n}{3+10^n}$, $b_n$ be ...
1
vote
1answer
83 views

Do we have $\sum_{n=1}^\infty 0=0$?

Simple question: Do we have $$\sum_{n=1}^\infty 0=0$$ ? Mathematically this seems obvious, but in practice I am very uncomfortable with this. Because nothing is perfect, so $0$ might not be quite ...
0
votes
1answer
12 views

$\phi$, and the uses of an alternate formula

I was trying to find the solution to the formula: $$x = \sum_{n=1}^\infty{x^{-n}}$$ I found it to be the golden ratio, or $\phi = \frac{1 + \sqrt{5}}{2}$. I do not know if this has already been ...
0
votes
1answer
50 views

Gauss Method to show [closed]

Could you please give me the way to solve this problem Using Gauss method to show if $x ≠ y + 1$ then $$ \sum_{i=0}^n (x-y)^i = \frac{(x-y)^{n+1}-1}{x-y-1}. $$
1
vote
1answer
27 views

Determine the radius of convergence of $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ (by the ratio test if possible)

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ Applying the ratio test gives $\frac{({n+1})^{({n+1})^{1/3}}}{n^{n^{1/3}}}z<1$. So ...
0
votes
0answers
15 views

Prove by induction: $E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$ Please just check what I've done

Prove by induction: $$E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$$ Let me show you what I've done. I think I'm right: $$n=1,$$ $$E[c_1U_1(X)] = c_1E[U_1(X)]$$ Okay so maybe this one looks ...