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

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25 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 ...
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1answer
66 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 $N$. ...
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1answer
33 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 ...
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0answers
44 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 ...
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2answers
32 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
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3answers
77 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 = ...
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2answers
71 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
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2answers
62 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 = ...
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1answer
21 views

What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function: $$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$
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4answers
95 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
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4answers
89 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 ...
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4answers
57 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 ...
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1answer
87 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 ...
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1answer
25 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 ...
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1answer
28 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 ...
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0answers
16 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 ...
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2answers
58 views

Value of the sum: $\binom{19}{0} - 1/2\binom{19}{1} +1/3\binom{19}{2} - 1/4\binom{19}{3} … -1/20\binom{19}{19}$?

How do I find the value of this sum? I tried taking out the equal binomial coefficients as factors but this didn't really simplify anything. I am stumped.
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2answers
60 views

How we can prove that: $\sum _{k=n}^n f\left(\frac{k}{n}\right)\le n\cdot \log(2)$?

$f:\left[0,1\right]\rightarrow R,\:f(x)=\frac{1}{1+x}$ and we have to show that $\sum_{k=n}^n f\left(\frac{k}{n}\right)\le n\cdot\log(2)$. What I know is just that: $n\cdot \log(2)=\int_0^1 ...
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0answers
15 views

Prove the max and min for Herfindahl-Hirschman Index

HHI(w) = $\sigma w_i^2$ where $\sigma $ is the summation sign, sorry but it will be great if anyone could tell me how to print the capitalized Greek letter. $w_i $ is the weights for each of N assets ...
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1answer
58 views

How simplify this sum?

I need help to simplify this sum : $$\sum_{i=0}^{x-1}\left(1-\dfrac{1}{2^i}\right)^{m-1}$$ Is it possible to simplify it ? Thank you.
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0answers
60 views

When can we assume this identity about series?

Suppose we are given that $$\sum_{i =a} ^b f (i)g (i) =\sum_{i =a} ^b f (i)h (i) \tag {1} $$ when can we conclude that $$\forall i =a,…,b:g(i)=h(i) \tag {2} $$ ? For example, if $g, h $ are both ...
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2answers
86 views

Find the sum of series $(1^2+1)1!+(2^2+1)2!+(3^2+1)3!+…+(n^2+1)n!$.

Find the sum of series $(1^2+1)1!+(2^2+1)2!+(3^2+1)3!+...+(n^2+1)n!$ I have found one method as i have shown in my answer below. But that form took me 30 mins to identify. ...
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1answer
87 views

evaluate trigonometric

To all the genius out there , here is a question about expresssing summation of hyperboilc functions : First of all, I've already proved that: $$\sinh(x + 1)- \sinh(x) = (-􀀀1 + \cosh (1)) \sinh(x) ...
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1answer
67 views

Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$?

Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$? I tried Faulhaber's formula and Bell number but couldn't ...
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4answers
139 views

How to prove that the value of $e$ is irrational without using the number $e$ itself [closed]

Recently I have tried to prove that the value of $e$ is irrational without using the number $e$ itself. I have seen that the number $e$ can be expressed as $$\lim_{n\to\infty}(1 + 1/n)^n;$$ however, ...
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1answer
39 views

Sum of absolute values is finite

Suppose $\lambda_m \in \mathbb{R}$ and suppose that $\sum_{m \in \mathbb{N}} \lvert i + \lambda_m \rvert^{-p} < \infty$. Then why does $$\sum_{(m,n) \in\mathbb{N}\times\mathbb{Z}} \left\lvert i \pm ...
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1answer
52 views

How we can show that $\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{2n}\right)$

We have $I_n=\int _{\pi }^{2\pi }\:\frac{\left|sin\left(nx\right)\right|}{x}\:dx,$ and we need to show that$\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}\right)$ I write ...
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2answers
60 views

radius of convergence of $\sum_{n=1}^\infty n!^2x^{n^2}$ [closed]

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n!^2x^{n^2}$
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1answer
46 views

Infinite summation of a trigonometric series

$\sum_{n=1}^{n=\infty}\sin(\frac{n\pi x}{L})\sin(\frac{n\pi y}{L})\surd(k^2+\frac{n^2 \pi^2}{L^2})$ I am trying to solve the above summation. I still could not figure out if this summation converges ...
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1answer
65 views

Simplifying Sum

How would one show that $$ \sum_{i=0}^n\binom{n}{i}(-1)^i\frac{1}{m+i+1}=\frac{n!m!}{(n+m+1)!} ? $$ Any hint would be appreciated. Note: I tried to recognize some known formula, but since I don't ...
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0answers
20 views

Help with simplification rules form sums and integrals.

IF you had a power series with summation notation and an integral what expressions would you be able to pull outside the integral and which would you be able to pull outside the sum.
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1answer
422 views

Simplification of a double sum involving partial sums of harmonic series

Could somebody explain the jump in the following equation? $$\frac{1}{n}\sum\limits_{i=1}^{n}\left[1 + \sum\limits_{j=i+1}^{n}\frac{1}{m}\right] = 1 + \frac{1}{nm}\sum\limits_{i=1}^{n}(n-i) $$
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0answers
34 views

Sum of combinatorics

I encounter the following sum of combinatorics and I can not seem to find the solution. Can anyone lend a hand? $\binom {n^c} {a} \binom {n^{c'}}{2a} + \binom {n^c} {a+1} \binom ...
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5answers
98 views

How to sum $\sum_{k=1}^n (k+1)(k)(k-1)$

Is there an intelligent way to do this sum without using sums of cubes and sums of squares? $$\sum_{k=1}^n (k+1)(k)(k-1)$$
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1answer
41 views

Taking the square of a finite series

I was reading something that involved taking the square of a sum $\sum_{i=1}^k(n_i-1)$. The author just presented the result. $$\left(\sum_{i=1}^k(n_i-1)\right)^2 = \sum_{i=1}^k(n_i^2-2n_i)+k + cross ...
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1answer
105 views

Find the value of $\sum_{m=1}^\infty tan ^ {-1}\frac{2m}{m^4+m^2+2}$

How to find value of this sum? $$\sum\limits_{m=1}^\infty \tan^{-1}\left(\frac{2m}{m^4+m^2+2}\right)$$ I can't understand how to simplify this. Should I use any trigonometric substitution to simplify ...
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1answer
33 views

Finding the radius of convergence for a Maclaurin series

I am required to find the radius of convergence for the function $$f(x) = 5x^3 - 6x^2 - 7x + 6$$ by first finding its Maclaurin series. I found the Maclaurin series to be $$ 6 - 7x -6x^2 + ...
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2answers
43 views

How to prove that a sequence converges

I am having some trouble understanding how I can show that a given series converges. I found a general explanation here that states: To prove that a sequence converges, it is sometimes easier to ...
3
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1answer
24 views

How to write product as a nested sums

I'm required to write $\prod\limits_{i=1}^n(1-j_{i})$ as a nested sum, where $j_{ll} \neq j_{k}$ if $u \neq k$. I undestand I'd get something in the form of ...
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2answers
35 views

Evaluate nested summation of a function

I'm trying to relearn summation simplification, I haven't touch math in a while. I'm having trouble simplify this nested summation down and I don't even know where to start. Could anyone please give ...
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1answer
69 views

How to show this equality holds?

I need to show that the following equality holds for any integer i,j,m,n and p where p is probability (0<=p<=1) Could you please help me?I think I should use hyper-geometric function but I could ...
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2answers
43 views

Help me understand this algorithm problem.

First, I'm not looking for an answer here, I'm just looking to understand the problem so that I can prove it. I'm trying to analyzing the worst case running time of an algorithm, and it must has ...
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2answers
41 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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47 views

how would i simplify this into an identity?

$$ B_{n,k}^{f\ln(g)} = B_{n,k}\left(\frac{d}{dx}[f(x)\ln(g(x))], \frac{d^2}{dx^2}[f(x) \ln(g(x)), \cdots, \frac{d^{n-k+1}}{dx^{n-k+1}}[f(x) \ln(g(x))]\right) $$ We know that: $$ B_{n,k}^{f\ln(g)} = ...
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2answers
73 views

Need to show following equality

I want to show that the following equality holds for any integer i,m, and n.I could not figure out how to show it analytically. Could you please help me? $$ \sum _{j=0}^n ...
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1answer
43 views

How do I extrapolate summation notation from a given series?

I am currently working on the power series for a homework assignment. I have to find the radius of convergence for the function $$\frac{10}{1+64x^2}$$ By setting up the $$\frac{1}{1+64x^2}$$ part ...
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1answer
23 views

Summation simplification explanation

I'm trying to understand summation for my algorithm course and it has been a while since I took discrete math. Could any body please explain how does summation simplification work from the problem ...
2
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1answer
61 views

How to choose a contour in order to use the residue theorem to sum up a series from Ryzhik?

I would like to know how to sum up to following series (from the Gradshteyn-Ryzhik tables): ...
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3answers
42 views

Finding the formula for summation of the series

I was just solving a competitive programming question, wherein I found out that a formula can be used for solving it efficiently. Problem statement: http://www.spoj.com/problems/TOHU/ I tried a lot to ...
10
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4answers
184 views

Calculate the infinite sum $\sum_{1}^\infty \frac{\log{n}}{2n-1}$

I would like to calculate an asymptotic expansion for the following infinite sum: $$\displaystyle \sum_{1}^N \frac{\log{n}}{2n-1}$$ when $N$ tends to $\infty$. I found that the asymptotic expansion ...