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

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2
votes
1answer
64 views

Sum over subsets of $\{1,2,\ldots,n\}$ of terms involving a product over that subset

I'm attempting to perform a sum, using products, using all possible combinations, in a function. How would I go about doing this? (I really need to find something that works.) For example, say I ...
5
votes
5answers
54 views

Demonstration of sum of powers of $2$ [duplicate]

Theorem : For every natural number $p$: $$\sum^p_{i=0} 2^i = 2^{p+1}-1$$ I trieed to demonstrate the theorem using induction Demonstration : $1)$ If we have $p=0$ then we get $2^0=2^{0+1}-1$ that is ...
3
votes
0answers
50 views

$\lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$

I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$? I thought to change this to an integral, namely ...
0
votes
0answers
19 views

Discrete Mathematics; Counting, Summations [duplicate]

Let n ≥ 1 be an integer. Prove that: $$ \sum\limits_{i=1}^n i(\frac{n}{i}) = n \bullet 2^{n-1} $$ I am not sure how to prove this, I think I need to use the derivative of $$(1 + x)^ n$$ any help ...
2
votes
1answer
18 views

Limit to zero of the $p$-norm

I have the $p$-norm defined as $$\|x\|_p=\left(\sum_{k=1}^n|x_k|^p\right)^\frac{1}{p}.$$ I am trying to find the limit as $p\to0^+$ of $\|x\|_p$. I've seen it defined as $\{x_k:x_k\neq0\}$. Why is ...
0
votes
1answer
30 views

Proof of a Combinatorial Summation

How many bitstrings of length $n+1$ have exactly $k+1$ many $1$s? Let $i$ be an integer with $k\leq i\leq n$. What is the number of bitstrings of length $n+1$ that have exactly $k+1$ many $1$s and in ...
0
votes
2answers
26 views

Binomial coefficient identity $\sum_{k=1}^n k {n \choose k } = n\cdot 2^{n-1}$ [duplicate]

I'm having a bit of problems proving the following: $$\sum_{k=1}^n k {n \choose k } = n\cdot 2^{n-1}$$ I always seem to get to the line: $2^{n-1} + 1 = 2^n$ which I know is untrue. Could anyone ...
3
votes
3answers
51 views

Proving that two summations are equivalent [duplicate]

Give a constructive proof to show that for all $n \geq 1$ , $\sum\limits_{i=1}^n i^3 = (\sum\limits_{i=1}^n i)^2$ Observe that $(n+1)^4 - n^4 = 4n^3 + 6n^2 + 4n + 1$ . Now, the two following ...
-1
votes
2answers
41 views

Help me understand how to take derivative of the PDF of X~binom(n,p) with respect to p.

This is the solution I was given. My questions: Why is it summed from k=1 to x. Shouldn't it be from k=1 to n? (If not, why not?) What is happening to the first term from line 1 to line 2? When we ...
1
vote
2answers
39 views

Proof of Binomial Formula Summation - Induction

Let $n\geq 1$ be an integer. Prove that $$ \sum_{k=1}^n k\binom{n}{k} = n\cdot 2^{n-1}. $$ Not sure how to go about doing this question. It says that finding the derivative of $(1+x)^n$ is useful. ...
-1
votes
0answers
21 views

Probability helps to evaluate a sum

Let's consider a sum $$\sum_{n=0}^{m} \binom {n+m} {n} \cdot 2^{-n}$$. How does this sum can be evaluated, considering the topics about probability? One of the solutions is written at the "Concrete ...
5
votes
0answers
66 views

Can we interchange the Integral and Summation when a limit is $\infty$?

I was trying to Evaluate the Integral: $$\Large{I=\int_1^{\infty} \frac{\ln x}{x^2+1} dx}$$ $$\color{#66f}{{\frac{1}{x^2+1} = \frac{1}{x^2\left(1+\frac{1}{x^2}\right)}=\frac{1}{x^2}\cdot ...
0
votes
1answer
28 views

How to use matlab for plotting functions that contain summations?

I am having a terrible time trying to figure out how to plot this function in matlab: $$\frac{1}{\pi} + \frac{1}{2}\sin(4t) - \frac{2}{\pi} \sum\limits_{k=2,4,6,8}\frac{\cos(4kt)}{k^2-1}$$ I am not ...
4
votes
1answer
56 views

Solve $(\alpha,\beta)$ for $\lim_{n\to\infty} \frac{\sqrt[n^2]{1!2!\cdots n!}}{n^\alpha} = \beta$

Find the ordered pair $(\alpha,\beta)$ with non-infinite $\beta \ne 0$ such that $$\lim_{n\to\infty} \frac{\sqrt[n^2]{1!2!\cdots n!}}{n^\alpha} = \beta$$ My approach: $$\ln (1!2!\cdots n!) = (n)\ln ...
5
votes
0answers
46 views

Heuristics of the sum of squared naturals $(1^2 + 2^2 + 3^2 \cdots + n^2)$

I'm new and this is my first question (though I've been lurking). English is not my native language. Studying on my own. I'm really interested in deriving the formula $1^{2} + 2^{2} + 3^{2} + \cdots+ ...
1
vote
1answer
51 views

Equivalent of the sum $\sum_{n=1}^\infty\frac{x^n}{\sqrt{n}}$

Let's consider $\displaystyle f(x)=\sum_{n=1}^\infty\frac{x^n}{\sqrt{n}}$. Where $f$ is defined, can we find a closed form for $f(x)$ ? What would be an equivalent of $f$ near $1^-$ ?
0
votes
1answer
31 views

How do I evaluate the sum $\sum_{i=1}^m \left[ \sum_{j=1}^n (i+j) \right]$?

I can't understand what this composition of summations practically says? Can someone explain it to me and if they can evaluate it? $$\sum_{i=1}^m \left[ \sum_{j=1}^n (i+j) \right]$$
1
vote
0answers
15 views

Sum of $p$th powers using polynomial interpolation

It is well known that the sum of the first $n$ $p$th-powers is polynomial in $n$ and is given by: $$ \sum_{k=1}^n k^p = \frac{1}{p+1} \sum_{j=0}^p (-1)^j {p+1 \choose j} B_j n^{p+1-j} $$ where $B_i$ ...
1
vote
1answer
19 views

Can someone explain why this summation is equal?

Can someone explain to me why this is equal? $$\sum_{i = 1}^n i = \sum_{i = 1}^n (n - i + 1) = \sum_{i = 0}^{n - 1} (n - i)$$
0
votes
3answers
32 views

Help with Infinite series + Integral Test + Improper Integrals

I am having some trouble with the infinite series $\displaystyle\sum_{n=2}^{\infty}\frac{1}{n\ln^2n}$ . I used the integral test and simplified it to $\int_{\ln 2}^b - \frac 1{\ln(n)}$ (implied ...
0
votes
0answers
14 views

A simple question about Delta Method's demonstration.

Suppose that $\sqrt{n}(X_n-\mu)\stackrel{D}{\longrightarrow}X$ and consider $g:\mathbb{R}\rightarrow\mathbb{R}$ a function such that first derivative $g'$ is continuous in a neighbourhood of $\mu$, ...
2
votes
1answer
30 views

Calculating $\sum_{k=0}^{n-1}\frac{1}{a+bk^2}$.

I want to calculate the following summation: $$\sum_{k=0}^{n-1}\frac{1}{a+bk^2}$$ Any hint how I can calculate this? Is there any kind of closed form for this summation?
0
votes
0answers
24 views

Is it possible to show the uniqueness of formula for solution?

The motivation to this question can be found in: Show that any sequence $(u_{n})$ must tends to infinity as $n→∞$ My question is: Is it possible to show the uniqueness of the formula for the ...
0
votes
1answer
22 views

Closed form solution of a summation

First off I have absolutely no clue what I'm doing, my notes given for this course do not explain anything and I'm not sure if I'm doing this properly so I'm looking for help and an explanation on how ...
0
votes
1answer
21 views

How to calculate the sum of a general series

In class we learned how to test the convergence of series and how to calculate the sums of arithmetic and geometric series (if they exist) but are there methods to actually calculate the values ...
3
votes
1answer
57 views

Proof that $\sum_{i=1}^n{1} = n$ for all $n \in \Bbb Z^+$

It seems obvious that $$\forall n \in \Bbb Z^+, \sum_{i=1}^n{1} = n $$ However, I'm having trouble coming up with a formal proof for this. Given a concrete number like $4$, we can say that ...
0
votes
2answers
30 views

proof by induction summation inequality

show by induction that: $$\sum_{i=1}^n i^2 = O(n^3)$$ what I have so far: $$\sum_{i=1}^n i^2 <= n^3$$ base case: for n=1 $$\sum_{i=1}^1 i^2 <= 1^3$$ ...
3
votes
4answers
154 views

Evaluate the sum $x + \frac{x^3}{3} + \frac{x^5}{5} + … $

Evaluate the sum $$x + \frac{x^3}{3} + \frac{x^5}{5} + ... $$ I was able to notice that: $$ \sum_{n=0}^\infty \frac{x^{2n-1}}{2n-1} = \sum_{n=0}^\infty \int x^{2n-2}dx = \lim_{N\to\infty} ...
0
votes
1answer
18 views

equality of two sums

Is $\sum _{t=1}^m \sum _{k=t}^m (a_{k+1}-a_k)b_t$ equal to $\sum _{k=1}^m \sum _{t=1}^k (a_{k+1}-a_k)b_t$? If not, then how to derive equations for temporal difference algorithm in machine learning: ...
3
votes
0answers
31 views

Is it possible to eliminate the inner sum to evaluate numerically?

Any hints on how to simplify the following double sum to be able to find the sum at least numerically? $$\sum_{n=2}^{\infty}\frac1{n(n^2-1)} \sum_{k=1}^\infty \frac{(k-1/n)^{2n-2}}{(k+1/n)^{2n+2}}$$ ...
0
votes
4answers
39 views

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

I need to find $$\sum_{r=1}^n q^{(r-1)k}$$ but I'm unsure on how to do this. Any help? I believe it may be something to do with a geometric progression but I'm not sure what to do with this. Thanks.
5
votes
0answers
54 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
1
vote
1answer
28 views

How to compute $\frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} w^{n^2} = \frac{1}{2}(1+i)\left(1+(-1)^N\right)$? [closed]

For $w=\exp[\frac{2\pi i}{N}]$, (i.e $N$th root of $1$.) \begin{align} \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} w^{n^2} = \frac{1}{2}(1+i)\left(1+(-1)^N\right) \end{align} How one can show this equation ...
2
votes
3answers
86 views

Simplify the sum $ \sum_{k=1}^{\infty} (\frac{1}{2})^kk $

I need some help simplifying this sum: $$ \sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^kk $$ I have a feeling it's some basic series thing that I'm forgetting, but I need help nonetheless.
2
votes
2answers
40 views

How to evaluate the sum $\sum_{j = i+1}^{n-1} j $?

How would I go about solving this summation? $$\sum\limits_{j = i+1}^{n-1} j $$ I'm trying to figure out how to solve this summation using the fact that $\sum\limits_{i = 1}^{k} i= ...
3
votes
1answer
42 views

Name of Inequality

Let $x_i, y_i$ be complex numbers for all $i$. Is there a name for the following inequality? $$\left| \sum_{i=1}^n x_i \right| \leq \sum_{j=1}^n |x_j| $$ In particular, is it a special case of this ...
2
votes
1answer
43 views

Asymptotics of $\sum_{\mathfrak{a}}\frac{n^{k-\epsilon}}{\mathfrak{N}\left(\mathfrak{a}\right)^{r\left(k-\epsilon\right)}}$

In this paper by Brian D. Sittinger, the following claim is made: For an algebraic number field $K$ with norm $\mathfrak{N}$, let $\epsilon=\left[K:\mathbb{Q}\right]^{-1}$. Then, taking the sum over ...
3
votes
2answers
35 views

Proving $1+\sum_{i=1}^n i (i!)=(n+1)!$ [duplicate]

How would you prove the following using induction. n is a non negative integer $$1+\sum_{i=1}^n i i!=(n+1)!$$ This be what I did base case let $n=3$ $$1+1+4+18=(3+1)!$$ $24=24$ Hypothesis step ...
2
votes
1answer
64 views

closed form for a double sum

How can I prove that $$\underset{k\geq1}{\sum}\left(\underset{m=-\infty}{\overset{\infty}{\sum}}\frac{\left(-1\right)^{m}}{\left(2k-1\right)^{2}+m^{2}}\right)=\frac{\pi\log\left(2\right)}{8}\,?$$I ...
0
votes
1answer
24 views

Divisibility of binomial coefficients

I have got this series of binomial coefficients - $${2n\choose 0}+3\times{2n\choose 2}+3^{2}\times{2n\choose 4}+\ldots +3^{n}\times{2n\choose 2n}$$ I have to prove this to be divisble by ...
0
votes
0answers
46 views

Proving that an infinite sum of irrationals is irrational

First of all, I know this question may be closed because it is off topic, but I do have a valid question. Problem: Is is possible to prove that an infinite sum of distinct and different irrational ...
0
votes
1answer
16 views

Simplifying this summation

I've been doing this question and I'm stuck! Each customer who enters Larry’s clothing store will, independently of every other customer, purchase a suit with probability p. Assume that N, the ...
2
votes
3answers
55 views

Proof by induction, binomial coefficient

I have to make the following proof: $${\sum\limits_{k=1}^n}{k}{n\choose k} = n2^{n-1}$$ Base case, $n = 1$: $${\sum\limits_{k=1}^{1}}{k}{1\choose k} = 1 = 1\cdot2^0=1$$ Inductive Hypothesis: for ...
0
votes
2answers
21 views

Minimum Value Given Average

I am struggling with this (very embarrassingly basic) minimization problem: I have the answer, which is great, but I'm oblivious as to how to go about solving this short of using calculus. The ...
0
votes
0answers
7 views

how to calculate the composite score / ranking or weight

I have $14$ parameters ( for e.g $A, B, C, D$, etc) which I have obtained rating of all from $1$ to $14$. I have calculated their final weights using rank order centroid which are absolute weights ...
2
votes
1answer
19 views

Prove the series converges uniformly at $[x_0, \infty)$

Let $\sum_{n=0}^\infty a_ne^{-\lambda_n x}$, where $0 < \lambda_n < \lambda_{n+1}$. It is given that the series converges at $x_0$. Prove that the series converges uniformly at $[x_0,\infty)$. ...
6
votes
1answer
54 views

How to prove $\lim_{n \to \infty}\frac{\pi}{2n+1}\sum_{k=1}^{n}(-1)^{k+1}\cot\frac{k\pi}{2n+1}=\ln2$

I am trying to prove the following: $$\lim_{n \to \infty}\frac{\pi}{2n+1}\sum_{k=1}^{n}(-1)^{k+1}\cot\frac{k\pi}{2n+1}=\ln2$$ I tried some values and it seems convincing. I wonder if this is a ...
1
vote
1answer
40 views

Lagrange's identity in the complex form

I am trying to show Lagrange's identity in the complex form; that is, $$ \Bigl\lvert\sum_{i = 1}^na_ib_i\Bigr\rvert^2 = \sum_{i = 1}^n\lvert a_i\rvert^2\sum_{i = 1}^n\lvert b_i\rvert^2 - \sum_{1\leq ...
8
votes
1answer
219 views

Identity in Number Theory Paper

In this paper by Jerry Hu, he defines the function $$f_{s,k,i}\left(u\right)=\prod_{p\mid u} \left(1-\frac{\sum_{m=i}^{k-1}{s \choose m}\left(p-1\right)^{k-1-m}}{\sum_{m=0}^{k-1}{s \choose ...
2
votes
2answers
59 views

Sum $\sum_{x=1}^n\sum _{y=1}^{x-1}\frac{1/2^x*1/2^y}{1/2^x+1/2^y}$

Is there a way to calculate following summation $\sum_{x=1}^n\sum _{y=1}^{x-1}\frac{1/2^x*1/2^y}{1/2^x+1/2^y}$ Can it be reduced to something simple?