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

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5
votes
0answers
70 views

Find the sum of the series below

Find the sum $$(1\cdot2)+(1\cdot3)+(1\cdot4)+\cdots+(1\cdot2015)+(2\cdot3)+(2\cdot4)+\cdots+(2\cdot2015)+\cdots+(2014\cdot2015)$$ What I have tried... We are looking for ...
0
votes
1answer
25 views

Help evaluating a partial sum with factorials and binomial coefficients

I come from a CS background and had to contend with a problem similar to this one. Essentially, I want a general-case estimate on how many rolls I'd have to make to land on the same number twice with ...
3
votes
4answers
68 views

Prove by induction: $\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=\frac{n!-1}{n!}$

Prove $$\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=\frac{n!-1}{n!}.$$ My problem with this is that it doesn't hold for the base case: $n=1$. This question is from the book "Abstract ...
0
votes
1answer
16 views

Summation of finite power seires

Is it possible to find a close form solution for $S_1$. $S_1$ is defined as follows: $S_1=\sum_{k=b}^{\infty}\frac{x^k}{k!}$ ; Where $0<x<b<\infty$ If $b=0$ then $S_2 = e^x$. But how do we ...
1
vote
1answer
16 views

Ensemble average of square of fluctuations proof

The ensemble average of a random variable $x$ is denoted as $X$ or $\left \langle x \right \rangle$, and is defined as: $$ X = \left \langle x \right \rangle = \lim_{N \to \infty} \frac{1}{N} ...
-3
votes
3answers
53 views

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

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
0
votes
0answers
43 views

Derivative of a summation series

Let $f_{n}(x)=x+(1-x)x^2+(1-x)(1-x^2)x^3+\cdots+(1-x)(1-x^2)\cdots(1-x^{n-1})x^n;\quad n\geq4$ then $f'(x)=\text{ ?}$ $$(A)\qquad (1-f_{n}(x)) \left(\sum\limits_{r=1}^n ...
6
votes
4answers
56 views

Proving $\sum_{i=1}^n 2^i = 2^{n+1} - 2$ using strong induction [duplicate]

I just started learning proof by induction in class, but got a problem requiring proof by strong induction. Here is the problem. Prove by strong induction: $$\sum_{i=1}^n 2^i = 2^{n+1} - 2$$ ...
-1
votes
1answer
48 views

How would I go about creating the summation formula

Consider the following program segment, where i, j , k, n, and counter are integer variables and the value of n (a positive integer) is set prior to this segment. ...
0
votes
1answer
30 views

How to get initial digits from a sum before adding?

How would you add digits in such a way that when having a result sum, the initial digits that were added could be extracted/calculated? For example, when having the following random digits 132748107 ...
2
votes
1answer
29 views

Closed form for a floor sum

I want to compute $$\sum_{i=a}^b \left\lfloor \frac{i}{k} \right\rfloor$$ Where $k < b < \infty$, and $a > 0$. I don't know where to begin (or if there's a closed form, for that matter), so ...
6
votes
2answers
89 views

Calculate $\sum_{i = 0}^{n}\ln\binom{n}{i}\Big/n^2$

Calculate $$\sum_{i = 0}^{n}\ln\binom{n}{i}\Big/n^2$$ I can only bound it as follows: $$\binom{n}{i}<\left(\dfrac{n\cdot e}{k}\right)^k$$ $$\sum_{i = ...
0
votes
1answer
55 views

Calculat sums of the form $\lim_{n\rightarrow\infty}\sum_{i=0}^{n}\left(\frac{i}{n}\right)^{f(n)}$

Problem: calculate the sums of the form: $$\lim_{n\rightarrow\infty}\sum_{i=0}^{n}\left(\dfrac{i}{n}\right)^{f(n)}$$ Inspiration: one problem lets us prove that ...
0
votes
2answers
34 views

Trying to understand why 2 times the sum of consecutive integers from 0 to n is equal to n times n+1

I am sorry if this question ends up being a duplicate, as I am having a bit of a challenge explaining it to myself well enough to know how to query it. There is a Facebook meme that has been ...
0
votes
0answers
51 views

Zeta function, how to solve a finite geomatry summation.

I wanted to solve the zeta function for an undifend period "$d$". So for every $d\ge2$. $$\zeta(-s)= \frac{1}{(d^{s+1}-1)}\sum_{m=1}^{\infty} \frac{1}{2^{m+1}}\sum^{m}_{j=1} ...
3
votes
0answers
64 views

Does there exist a closed form?

I wish to find a closed form for $\sum_{i=1}^n\frac{1}{i}$. does it exist? If so, what is it? I cannot arrive at one using any methods I am aware of.
0
votes
0answers
12 views

Double Sum of Series Unchanged When Each Term Scaled

Can the following ever hold: $$\sum_{i}\sum_{j}(-1)^ja_{i,j}=\sum_{i}i(b_{i})\sum_{j}(-1)^ja_{i,j}$$ where $b_{i}>1/i$ for all $i$? What if you tack on the fact that ...
0
votes
2answers
63 views

Help needed with the integral of an infinite series

Can you please help me with the integral of this series? I came across it in a signal processing paper and haven't been able to figure out the solution myself. $$ ...
9
votes
2answers
209 views

Help with binomial identity

In my work, I was led to the following identity. I would be grateful if someone could provide an easy proof. Suppose $n, d, k \in \mathbb{Z}$, and $d \geq 0$. $$ \sum_{j = 0}^d (-1)^{d-j} \cdot ...
-2
votes
1answer
45 views

Limit and summation. [on hold]

I find This limit and I want to get the value of it $$\displaystyle\lim_{n\to\infty} \displaystyle\sum_{i=1}^{n} \frac{7i^{3}+i^{2}+3i +1}{n^{4}+i+5} =? $$ I tried Riemann susummation t still can't ...
-3
votes
1answer
31 views

summation of some random numbers [on hold]

Use 1-20 numbers , pick six numbers and those six numbers summation should be 20 . conditions: the numbers cannot be repeated , and numbers should with in the range 1-20, perform addition only
2
votes
2answers
22 views

Convegence of $\sum_{i\in J}a_i$ implies that index set is countable

Let $J$ be a uncountable set and $\{a_i\}_{i\in J}$ be a set of non-negative real numbers. Prove that $\sum_{i\in J}a_i<\infty$ implies that there is a countable set $H\subset J$ such that $a_i=0$ ...
2
votes
1answer
50 views

Infinite Telescoping Sum: $\sum_{i=1}^{\infty} (X_i - X_{i-1})=$?

Let $(X_i)_{i \geq 0}$ be any countable sequence of numbers and suppose that a limit exists, so $$\lim_{i \rightarrow \infty} X_i = x.$$ Consider $\sum_{i=1}^{\infty} (X_i - X_{i-1})$. Is this ...
0
votes
0answers
18 views

Notation sumation confusion

I am reading paper about additive schwarz preconditioner, where following notation is used in order to obtain matrix C $$C_i = \sum_k (I^k B^k (P^k u_i)R^k)$$ . My question is, what's dimension of ...
1
vote
2answers
49 views

Summation of a logarithmic series for $\ln(2(r^2 - 1)/r^2)$

Given that $$\sum_{r=2}^{n}\ln\frac{r^2-1}{r^2}=\ln\frac{n+1}{2n}$$ for $n >1$. Express $$\sum_{r=32}^{62}{\ln\frac{2(r^2-1)}{r^2}}$$ as $$A\ln 2 + B\ln3 + C\ln7$$ where $A$, $B$, $C$ are positive ...
4
votes
0answers
59 views

How is it that $\pi$ appears in so many formulas that seem to be in no way geometric. [duplicate]

When I first saw: $$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}\mp\cdots.$$ I was puzzled that such an expression could have anything to do with circles. There are tons more and ...
3
votes
0answers
42 views

Reorder this series to change its sum [duplicate]

If in the series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\cdots$ the order of the terms be altered, so that the ratio of the number of positive terms to the number of negative terms in the first $n$ ...
3
votes
2answers
104 views

Algebraic proof that $\sum\limits_{i=0}^n \binom{i}{k} = \binom{n + 1}{k + 1}$

I'm looking for an algebraic proof of this identity for $n, k \in \mathbb{N}$: $$\sum\limits_{i=0}^n \binom{i}{k} = \binom{n + 1}{k + 1}$$ So far, I've turned the left hand side of the equality into ...
3
votes
3answers
51 views

Proving $\sum_{i=1}^n\frac{1}{i(i+1)(i+2)}=\frac{n(n+3)}{4(n+1)(n+2)}$ for $n\geq 1$ by mathematical induction

Prove using mathematical induction that $$\frac{1}{1\cdot 2\cdot 3} + \frac{1}{2\cdot 3\cdot 4} + \cdots + \frac{1}{n(n+1)(n+2)} = \frac{n(n+3)}{4(n+1)(n+2)}.$$ I tried taking $n=k$, so it makes ...
4
votes
2answers
143 views

Some infinite series with Fibonacci numbers

An interesting problem is to prove that: $$ \sum_{n=1}^\infty \frac{F_{2n}}{n^2 \binom{2n}{n}}=\frac{4\pi^2}{25 \sqrt 5}. $$ I know the proof, which uses the fact that ...
3
votes
4answers
112 views

Proving that $1\cdot3+3\cdot5+5\cdot7+\cdots+(2n-1)(2n+1)={n(4n^2+6n-1) \over 3}$ by induction for $n\geq 1$

Prove using mathematical induction that $$1\cdot3+3\cdot5+5\cdot7+\cdots+(2n-1)(2n+1)= {n(4n^2+6n-1) \over 3}.$$ Step 1: If we assume that the equation is true for a natural number, $n=k$, ...
3
votes
1answer
33 views

An approach to approximating the harmonic series.

I would like to get help on the last step to approximating the harmonic series, here is my work: Consider the equation: $$f(x+1)-f(x)=g(x)$$ Through iteration one can come up with the solution: ...
11
votes
5answers
1k views

Sum of an infinite series $(1 - \frac 12) + (\frac 12 - \frac 13) + \cdots$ - not geometric series?

I'm a bit confused as to this problem: Consider the infinite series: $$\left(1 - \frac 12\right) + \left(\frac 12 - \frac 13\right) + \left(\frac 13 - \frac 14\right) \cdots$$ a) Find the sum $S_n$ ...
1
vote
0answers
70 views

A Summation Challenge

I am trying to understand the solution of problem from its editorial by djdolls' answer,I am not able to understand a particulare step which is as follows: $$S(n)=\sum_0^D (-1)^i \cdot ...
2
votes
2answers
102 views

Prove $\sum\limits_{i=0}^{n}\binom{n+i}{i}=\binom{2n+1}{n+1}$ [duplicate]

I'm trying to prove this algebraically: $$\sum\limits_{i=0}^{n}\dbinom{n+i}{i}=\dbinom{2n+1}{n+1}$$ Unfortunately I've been stuck for quite a while. Here's what I've tried so far: Turning ...
-6
votes
3answers
69 views

Summing odd numbers [closed]

$$1+3+5+7+ \cdots + (2n-1) = \, ??$$ Can you help me?
3
votes
0answers
100 views

Using a visual “proof” to show that $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n =1$

The "proof without words" that $\sum_{n=1}^{\infty} \left(\frac 12 \right)^n =1$ is fairly well known: But why can't we apply the exact same logic to $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n$ ...
0
votes
1answer
32 views

Converging Geometric Series of a a Bouncing Ball

A rubber ball was dropped from a height of 36m. and each time its strikes the ground it rebounds to a height of 2/3 from which it last fell. Find the total distance traveled by the ball before it ...
1
vote
1answer
31 views

Evaluate products and sums

Did I evaluate the following terms correctly? Does the set notation in example b) allow me to chose the order of the terms? $$ a) \sum_{i=1}^6 ix^{i+1} = x^2+2x^3+3x^4+4x^5+5x^6+6x^7 \\ b) \prod_{i ...
5
votes
3answers
74 views

If $\lim_{n\to\infty}\frac{1^a+2^a+…+n^a}{(n+1)^{a-1}.((na+1)+(na+2)+…+(na+n))}=\frac{1}{60}$, Find the value of a

If $$\lim_{n\to\infty}\frac{1^a+2^a+...+n^a}{(n+1)^{a-1}\cdot((na+1)+(na+2)+...+(na+n))}=\frac{1}{60}$$ Find the value of a. Attempt: I solved it using two methods each giving me different answers. ...
5
votes
1answer
186 views

$\sum_{i=1}^n \frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod _{j=1}^nx_j}} \ge 1$, for all $x_i>0$

Can you help with the following inequality? I found it experimentally. Prove that, for all $x_1,x_2,\ldots,x_n>0$, $$\sum_{i=1}^n\frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod _{j=1}^nx_j}} \ge ...
1
vote
2answers
43 views

Finding a sum involving roots of a quadratic equation

If $\alpha,\beta$ are roots of the equation $x^2-2x-7=0$ and $$S_r=\left(\frac{r}{\alpha ^r}+\frac{r}{\beta ^r}\right)$$ then find the value of $$\lim _{n \to \infty} \sum _{r=1} ^n S_r$$ I am unable ...
5
votes
1answer
68 views

How to compute the sum $\displaystyle\sum_{n=0}^{\infty}nP_{n}$

Following Problem is from probability theory:Define $G(n),P(n)\ge 0,n\in\mathbb{N}$,and such $$\begin{cases}G(n)=e^{-\lambda}\cdot\dfrac{\lambda^n}{n!},\lambda>0\\ ...
0
votes
1answer
166 views

Algebra question about inequalities [closed]

Let $n>0$ and let there be two positive integers $x,y$ such that $x^n+y^n=1$ Prove, $$\left(\sum_{k=1}^{n} \frac{1+x^{2k}}{1+x^{4k}}\right)\left(\sum_{k=1}^{n} ...
0
votes
2answers
40 views

Identity with complex numbers related to the Cauchy-Schwarz inequality

I have this equation $ a_j,b_j\in \mathbb{C} , j=1,2,...,n$ $$ \left| \sum\limits_{j=1}^n a_jb_j \right|^2 = \sum\limits_{j=1}^n |a_j|^2 \sum\limits_{j=1}^n |b_j|^2 -\sum_{1\leq i \leq j \leq n} ...
0
votes
1answer
30 views

Can $\dfrac{b_0}{a_0} + \dfrac{b_1}{a_1} + \dfrac{b_2}{a_2} + \dfrac{b_3}{a_3} + … + \dfrac{b_n}{a_n}$ be represented as …

Is this correct? (Last step $\rightarrow$ After taking L.C.M.) $\large \dfrac{b_0}{a_0} + \dfrac{b_1}{a_1} + \dfrac{b_2}{a_2} + \dfrac{b_3}{a_3} + ... + \dfrac{b_n}{a_n} = \sum\limits_{k=0}^{n} ...
1
vote
1answer
45 views

What does the sum of the reciprocals of composites run along?

This is fairly straight forward: $$\sum_{p\space\text{prime}}^x \frac{1}{p_x} \sim \ln(\ln(x))$$ And if $$\sum_{c\space \text{composite}}^x \frac{1}{c_x}\sim f(x)$$ Then what is $f(x)$?
3
votes
4answers
156 views

Prove this inequality: $\frac n2 \le \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+…+\frac1{2^n - 1} \le n$

$\dfrac{n}{2} \le \dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2^n - 1} \le n $ I've Tried for hours but didn't got any striking idea. I don't have any efforts to show rather than induction. ...
0
votes
1answer
40 views

Convergent conjecture: What is the proof?

Lets say that $\def\nn{\mathbb{N}}$$\def\rr{\mathbb{R}}$$K : \nn \to \rr$ and $\displaystyle \sum_{i=1}^\infty \frac{K(i)}{K(i+1)}$ is a convergent sum. My conjecture is that the function $K$ must be ...
4
votes
2answers
206 views

Finding a recurrence for a sum

I am trying to implement the following sum using a programming language: $$\sum_{i=1}^N a^i i^r$$ where $N$, $a$ and $r$ are integers. The problem is, I cannot find a suitable way to do this. ...