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

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4
votes
3answers
66 views

How to derive the closed form of the sum of $kr^k$

$$ \sum_{k=0}^{n}kr^k = r\frac{1-(n+1)r^n + nr^{n+1}}{ (1 - r)^2 } $$ How to derive it? I read about some finite calculus, and i understand how to tackle sums of $x^2$, $x^3$, etc.. But I don't know ...
4
votes
2answers
115 views

Is there any closed form for the finite sum $1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{n}?$ [duplicate]

I know that the infinite summation $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}+...$$ is divergent and also the sequence $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}-\ln ...
1
vote
1answer
32 views

Summation of numbers $ (n+1)(n+2)\cdots(n+m)$ over $n$

Verify that if $V_n = n(n+1)(n+2)\cdots(n+m)$ then $$V_{n+1} - V_n = (m+1)(n+1)(n+2)\cdots(n+m)$$ Given now that $U_n = (n+1)(n+2)\cdots(n+m)$ find sum of series $U_n$ from $N$ to $1$ in terms ...
1
vote
1answer
54 views

Derive the formula for the sum of the first $n$ squares using derivatives and integrals

I wanted to prove the formula for sum of squares without using induction and thought using derivatives would be the easiest approach ...
0
votes
1answer
52 views

Summation of series of $2/(r-1)(r+1)$ using the method of differences

Verify the identity $$\frac{2r-1}{r(r-1)}-\frac{2r+1}{r(r+1)}=\frac{2}{(r-1)(r+1)}$$ Hence, using the method of differences, prove that ...
0
votes
2answers
25 views

Identity involving trigonometric sum

I have to prove that $$\overset{N}{\underset{n=-N}{\sum}} \left(N-\left|n\right|\right)e^{2\pi inx}=\left|\overset{N}{\underset{n=1}{\sum}} e^{2\pi inx}\right|^{2}=\left(\frac{\sin\left(N\pi ...
0
votes
2answers
27 views

Why is this term $=1$

Can you tell me why $$\frac{1}{r} \sum_{k=0}^{r-1} R_N(x^k) \sum_{s=0}^{r-1} e^{\frac{-2 \pi i s k}{r}}=1?$$ Here $R_N(x^k)$ is the remainder of $x^k$ Modulo $N$. When I entered the last sum in ...
1
vote
2answers
38 views

Naive proof that $\sum_{n=1}^{N-1}\cos(2\pi\frac{n}{N})=-1$ [duplicate]

As part of a larger proof, I must show that: $$\sum_{n=1}^{N-1}\cos(2\pi\frac{n}{N})=-1$$ I have thought about this but can't figure out how to get my hands on the value since I don't know any ...
0
votes
3answers
45 views

Find the sum of this series

May I know how I should go about finding the sum of this series? $\displaystyle\sum_{n=1}^\infty$ $\dfrac{n}{2^{n-1}}$ I am really stuck. Thanks!
1
vote
2answers
49 views

Finding a formula for $1+\sum_{j=1}^n(j!)\cdot j$ using induction

I need help with finding the formula and proving it by induction. Am stuck, but the professor says we should know this by now.
0
votes
0answers
22 views

The probability that uniformly distributed integers sum to a given integer

A recent CTF had a problem involving the summation of randomly distributed integers. Specifically: Consider a set $\{X_m\}$ of $M$ integers uniformly selected (with replacement) from the set of ...
0
votes
2answers
35 views

Simplifying modulus expressions and an unknown expression? discrete math

I have a few questions below that I need help with a) I don't really understand what that symbol means and how to solve it b) How do u simplify this without a calculator c) I got 2^-r = 0, iss this ...
3
votes
5answers
111 views

Computing $\sum_{i=0}^{\infty}\frac{i}{2^{i+1}}$

I came across this while trying to solve Google's boys & girls problem, and although I know now it's not the right approach to take, I'm still interested in summing ...
2
votes
1answer
64 views

What is the correct way to write sum from 47 to 54 so it means 404?

Does the above image correctly mean "This is a 404 error?" That is, does this $$ \sum_{i = 47}^{54} $$ mean 404? Or should it just be this, $$ \sum_{47}^{54} $$ without the equals?
2
votes
2answers
55 views

Show that $H_i=H_{n-i}$ and $\sum H_i=1$

We define $$H_i=\frac{1}{n}\frac{(-1)^{n-1}}{i!(n-1)!}\int_{0}^{n}\prod_{j=0,j\neq i}^{n}(x-j)dx$$ This is called the Newton-Cotes coefficient. Here is the exercise: First, convince yourself that ...
2
votes
0answers
23 views

Sum of degrees of irreducible complex characters for certain groups

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant do determine the dimension of a maximal torus in the group algebra. I have ...
4
votes
3answers
83 views

How to find the sum of sequence $ 1+4+4^2+\cdots+4^{X+Y} $?

I see the following sequence and it's: $$h=1+4+4^2+\cdots+4^{X+Y}=\frac{4^{X+Y+1}-1}{4-1}$$ how we get this sequence? I know this is a primary question but I confused :)
2
votes
2answers
55 views

Alternating sum of binomial coefficients is equal to zero [duplicate]

Prove without using induction that the following formula:$$\sum_{k=0}^n (-1)^k\binom{n}{k}=0$$ is valid for every $n\ge1$. Progress For each odd $n$ we can use the ...
2
votes
2answers
70 views

How to prove $n! > n^a$ for all $a\in \mathbb{R}$ (for sufficiently large $n$)?

I've encountered a proof which claims $n! > n^2$ for sufficiently large $n$. I tried using induction to prove it for an arbitrary $a$: $n! > n^a$. Lets assume the claim is true for $n$: $n! ...
0
votes
1answer
36 views

Prove the serie is bounded by $a_{m+1}$ for all $m\in \Bbb{N}$

Let $a_n$ a monotone sequence approaches $0$. Show that for all $m\in\Bbb{N}$: $$ 0 < (-1)^m\sum\limits_{n=m+1}^{\infty} (-1)^{n+1} {a_n} < a_{m+1} $$ I wanna focus on the RHS inequality: ...
2
votes
1answer
43 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
3
votes
3answers
83 views

Evaluating the sum $1\cdot 10^1 + 2\cdot 10^2 + 3\cdot 10^3 + \dots + n\cdot 10^n$

How can I calculate $$1\cdot 10^1 + 2\cdot 10^2 + 3\cdot 10^3 + 4\cdot 10^4+\dots + n\cdot 10^n$$ as a expression, with a proof so I could actually understand it if possible?
-1
votes
1answer
56 views

Proof by Induction [Number Theory by George E. Andrews 1-1 #2] [duplicate]

I am to use mathematical induction to prove that: $$1^3 + 2^3 + 3^3 + \cdots + n^3 = (1 + 2 + 3 + \cdots + n)^2 $$
0
votes
1answer
34 views

Value of an iterated sum

I am interested in the number of function evaluations required to numerically evaluate an iterated integral of the form $$ \int_0^t \int_{t_1}^t \cdots \int_{t_{n-1}}^t f(t_1,\ldots,t_n) dt_n\cdots ...
0
votes
1answer
30 views

Simplifying the sum $\sum_{i=1}^n \sum_{j=i+1}^n({(x_i-x_j)}^2+{(y_i-y_j)}^2)$

I am trying to evaluate the sum here , $$\sum_{i=1}^n \sum_{j=i+1}^n({(x_i-x_j)}^2+{(y_i-y_j)}^2)$$ How do this sum can be simplified to $$n\sum_{i=1}^n({x_i}^2+{y_i}^2) - ...
4
votes
4answers
72 views

Find the sum of the multiples of $3$ and $5$ below $709$?

I just cant figure this question out: Find the sum of the multiples of $3$ or $5$ under $709$ For example, if we list all the natural numbers below $10$ that are multiples of $3$ or $5$, we get $3$, ...
0
votes
0answers
42 views

Summation of cosines

$n$ is any positive integer and $j=0,1,...,[n/2]$, where $[]$ denotes the greatest integer function. How do you prove that for $j \notin\{0,\frac{1}{2}\}$ $$\sum_{t=1}^n \cos^2\left(\frac{2\pi t ...
1
vote
3answers
132 views

Approximate summation of the given equation

I have been trying from an hour to approximate the value of $M$ in the equation given below. $$ M = \sum\limits_{i=1}^n\left(\sum\limits_{j=1}^n\left(\sqrt{ i^2 + j^2 }\right)\right) $$ One thing I ...
7
votes
2answers
138 views

Sum of $1+\frac{1}{2}+\frac{1\cdot2}{2\cdot5}+\frac{1\cdot 2\cdot 3}{2\cdot 5\cdot 8}+\cdots$

I am trying to find out the sum of this $$1+\frac{1}{2}+\frac{1\cdot2}{2\cdot5}+\frac{1\cdot 2\cdot 3}{2\cdot 5\cdot 8}+\frac{1\cdot 2\cdot 3\cdot 4}{2\cdot 5\cdot 8\cdot 11}+\cdots$$. I tried with ...
3
votes
2answers
43 views

A sum of difference of floors

I have the sum ( $M$ is any integer $> 1$ ): $$ \sum_{h = 1}^{M}\left(\,\left\lfloor\, 2M + 1 \over h\,\right\rfloor -\left\lfloor\, 2M \over h\,\right\rfloor\,\right) $$ and looking for a way to ...
4
votes
6answers
722 views

What is the limit of the following sum

$$\lim_{n\to\infty}\sum_{k=1}^n \ln\Big(1+\frac{k}{n^2}\Big)$$ According to me, the answer is $0$. I'm curious as to what answers might others come up with, as well as the method of reasoning.
1
vote
1answer
49 views

Prove the identity involving summation and Stirling numbers of the second kind

Prove the identity $$(e^z-1)^m=m!\sum_{n}^{}{n \brace m}\frac{z^n}{n!}$$ $n\brace m$ stands for Stirling numbers of the second kind. I'm not really sure if $z$ is some special number or just an ...
0
votes
2answers
29 views

Average length of a bitstring

I am trying to compute the average length of a bit string from all bit strings of $\{0,1\}^n$. By length n I mean a bit string of length n where the most significant bit is 1. I know there is $2^0$ ...
2
votes
2answers
56 views

I need a set that enables me to identify specific containing elements by any summation of any of its subsets (see example to understand)

My question is more practically understood by example. I need a set A that behaves like the one below: Set A: {1,3,5} Set B (all subsets of A): {1}, {3}, {5}, {1,3}, {1,5}, {3,5}, {1,3,5} Set C ...
1
vote
2answers
40 views

Problem involving summation and binomial coefficient

I have been fighting with this but I'm really not getting anywhere. $$\sum_{0\leq2k\leq n}\binom{n}{2k}2^k\equiv0\pmod 3$$ $$iff$$ $$n\equiv2\pmod 4$$ Hint: Consider ...
3
votes
2answers
147 views

Evaluation of the sum $\sum_{i=1}^{\lfloor na \rfloor} \left \lfloor ia \right \rfloor $

Let $a$ be a positive proper fraction and $n$ is any integer then evaluate the following sum, $$\sum_{i=1}^{\left \lfloor na \right \rfloor\atop} \left \lfloor ia \right \rfloor $$ I think that ...
7
votes
1answer
117 views

Showing that $\sum_{i=1}^n \frac{1}{|x-p_i|} \leq 8n \left( 1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n-1} \right)$

I'm taking a summer analysis course and preparing for our final exam later this week. Our professor gave us the following problem on our mock exam, and I can't seem to get anywhere on it. Does anyone ...
3
votes
0answers
36 views

Solving the recurrence $T(n)=4T(\frac{\sqrt{n}}{3})+ \log^2n$ [closed]

How we calculate the answer of following recurrence? $$T(n)=4T\left(\frac{\sqrt{n}}{3}\right)+ \log^2n.$$ Any nice solution would be highly appreciated.
3
votes
1answer
73 views

How should I prove that: $\sum_{i=1} ^{n}(\sin(\frac{i\pi}{n}))^2=\frac{n}{2}$

$$\sum_{i=1} ^{n}\Big(\sin\big(\frac{i\pi}{n}\big)\Big)^2=\frac{n}{2}$$ An interesting conclusion and checked for validity...holds for $n\geq 2$, but yet do not know how to prove it. Are there any ...
0
votes
1answer
52 views

find the sum of the series

If $a_1, a_2, \ldots, a_n$ are in arithmetic progression whose common difference is $d$,then find the sum: $$\sin(d) \cdot \left(\csc(a_1)\csc (a_2)+\csc(a_2)\csc (a_3)+\ldots+\csc(a_{n-1})\csc(a_n) ...
1
vote
2answers
44 views

How to show the identity relating to Matrix

Suppose that $$ A=\begin{bmatrix}a_{11}&a_{21}\\a_{21}&a_{22}\end{bmatrix}, \ \ B=\begin{bmatrix}d&-1\\1&0\end{bmatrix}. $$ and $$A=B^N$$ Show that $$a_{11}=\sum_{i=0}^{[N/2]}(-1)^i ...
1
vote
3answers
82 views

Finding the sum of $3+4\cdot 3+4^2\cdot 3+\dots +4^{\log n-1} \cdot 3$

I see this: $$A=3+4\cdot 3+4^2\cdot 3+\dots +4^{\log n-1} \cdot 3=3\cdot ([4^{\log n}-1]/3)=n^2-1$$ The base of logarithm is $2$, and $n$ is $2,4,8,\dots$ Anyone could describe me how this sum was ...
13
votes
1answer
172 views

A Gamma limit $\lim_{n\rightarrow+\infty}\sum_{k=1}^n \displaystyle \left( \Gamma\bigl(\frac{k}{n}\bigr)\right)^{-k}=\frac{e^\gamma}{e^\gamma-1}$

Show that $$\lim_{n\rightarrow+\infty}\sum_{k=1}^n \displaystyle \left( \Gamma\bigl(\frac{k}{n}\bigr)\right)^{-k}=\frac{e^\gamma}{e^\gamma-1}$$ where $\gamma$ is the Euler-Mascheroni Constant. ...
2
votes
1answer
112 views

Find sum of $n$ terms of the series $12+14+24+58+164+\cdots$

Find sum of $n$ terms: $12+14+24+58+164+\cdots$ I have tried my best but could not proceed
2
votes
3answers
61 views

Find partial sums of the series $12+105+1008+10011+\dots$

Find the sum of $n$ terms of this series- $$12+105+1008+10011+.....$$ I did not understand that how should I proceed with this problem.
4
votes
2answers
70 views

Summation of general series

One of the problems in Donald Knuth's Art of Programming is phrased as follows: Find and prove a simple formula for the sum $\sum\limits_{n=0}^k\frac{(-1)^n(2n+1)^3}{(2n+1)^4+4}$ I have very little ...
1
vote
1answer
52 views

Sum of the trigonometric series

I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far: \begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} ...
3
votes
3answers
120 views

Evaluation of a sum of $(-1)^{k} {n \choose k} {2n-2k \choose n+1}$

I have some question about the paper of which name is Spanning trees: Let me count the ways. The question concerns about $\sum_{k=0}^{\lfloor\frac{n-1}{2} \rfloor} (-1)^{k} {n \choose k} {2n-2k ...
3
votes
2answers
55 views

Which natural numbers can be represented as a sum of natural numbers raised to different powers?

Waring's problem asks about natural numbers that can be represented as a sum of natural numbers all raised to the same power $k$. I'm wondering which natural numbers can be represented as a sum of ...
2
votes
2answers
69 views

Why does this sum equal to (4^n -1)

How do I get to this solution? $\sum _{k=1}^n\left(\binom nk 3^{n-k}\right)=\left(4^n-1\right)$ I believe it's connected to this, which I know is true: $\sum \:_{k=1}^n\binom nk=2^n-1$