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

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6
votes
2answers
107 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
56 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
37 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
63 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} ...
2
votes
1answer
81 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
69 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. $$ ...
11
votes
2answers
234 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
2answers
29 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
51 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 ...
1
vote
2answers
51 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
60 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
110 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
53 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
159 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
37 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: ...
12
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
76 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
105 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
101 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
34 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
32 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
77 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
203 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
46 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
171 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
42 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
32 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 $f(x)$?
4
votes
4answers
198 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
41 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
210 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. ...
2
votes
3answers
344 views

Why does this sum converge?

I know that the following sum converges to 2 via WolframAlpha, but I am not sure why. $$\sum_{k=1}^\infty k \left[\frac{2}{k} - \frac{4}{k+1} + \frac{2}{k+2}\right] = 2$$ WolframAlpha gives the ...
1
vote
1answer
58 views

Doubling sequences of the cyclic decimal parts of the fraction numbers

Is there any theory, why and when doubling sequences of the decimal part of the fraction numbers occur? Take for example these small numbers: ...
0
votes
0answers
44 views

Sum involving binomial $\sum_{k=0}^{n} \binom{3n}{3k}$ [duplicate]

The main question is to evaluate: $$\sum_{k=0}^{n} \binom{3n}{3k}$$ There is a standard technique but I cannot split the sums apart and then add them together. Could you help with this step?
1
vote
2answers
159 views

Proof of the reciprocal of all semiprimes diverging?

$$\sum_{\text{semi-primes}}\frac{1}{s}=\frac{1}{4}+\frac{1}{6}+\frac{1}{9}+\frac{1}{10}\cdots$$ I almost positive that this sum diverges, but I would really like to see a very thorough proof. Thank ...
0
votes
2answers
29 views

How to compute the following double sum $\sum_{z_1,z_2 \in [-N,..N]: z_1 \neq z_2} z_1^2-2z_1z_2$

How to compute the following double summation \begin{align*} \sum_{z_1,z_2 \in [-N,..N]: z_1 \neq z_2} z_1^2-2z_1z_2 \end{align*} I was thinking I can do the following \begin{align*} \sum_{z_1,z_2 ...
0
votes
0answers
51 views

Prove FTC using limit of summation

It is not hard to show $$\int_a^bx^2\,dx=\lim_{n\to\infty}\left[\frac{b-a}{n}\sum_{k=1}^n\left(a+(b-a)\frac kn\right)^2\right]=\frac{b^3}{3}-\frac{a^3}{3}.$$ With some effort one can also show ...
0
votes
2answers
54 views

Summation of a series of Positive Prime numbers

Is there a way to find the sum for a set of positive prime numbers (e.g., the first $25$ prime numbers) without just arbitrarily adding them up shorthand?
4
votes
1answer
49 views

Recurrence relation in terms of another sequence

How do I solve a recurrence of the form $$nd^{n-1}a_n+a_{n+1}d^{n+1}=b_n$$ for $a_n$, where $b_n$ is another (known) sequence and $d$ is a constant? My only idea was to use a generating function and ...
1
vote
1answer
37 views

Eulers proof sum of natural numbers

I've to recheck Eulers proof of the sum of the natural numbers, but I dont now exactly what it is? It has something to do with the $\zeta(s)$? Thanks in advance
1
vote
1answer
32 views

Clever way to simplify sum?

Is there a clever way to rewrite the sum $$\sum_{i=2}^{n} (x_i-x_{i-1})\left(\frac{(x_i-x_{i-1})}{2}-x_i \right) ?$$ I haven't been able to come up with anything useful thus far.
6
votes
2answers
98 views

Do the sum of all prime reciprocals with the digit $3$ converge or diverge?

$$\frac{1}{3}+\frac{1}{13}+\frac{1}{23}+\frac{1}{31}+\frac{1}{37}+\frac{1}{43}\cdots$$ Intuitively, I feel that this sum converges, but I really don't know why, (or if I am correct). Can I have a ...
0
votes
0answers
48 views

Question on how to manipulate terms in this expression

sorry for the vague title, i dont know how else to express what i mean with this question. But what i need to do is find out which terms on the RHS of the expression are constants. It is clear that it ...
1
vote
3answers
36 views

Switching the order of summations of a certain function

I am looking to switch the order of the summations of the following function: $$ \lambda = -\sum_{c=1}^{n-1} \sum_{k=c}^n {k \choose c} \frac{(-1)^k}{k!} f^{k-c}U(-c,k-2c+1,-f)\phi(n,k) $$ I don't ...
0
votes
0answers
55 views

Finding closed form for this summation

recently i have beeen asking alot of questions about summations, But this one is actually quite interesting: $$ \sum_{j=k}^n j! 2^{k-2j} \left({2j-k-1 \choose j-1} - {2j-k-1 \choose j}\right){n \brack ...