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

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0
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1answer
16 views

Finding $n$ from the cumulative sum of the serie where $SUM(n) < \Pi < SUM(n+1)$

I have a serie of numbers: $$S = {1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/20, 1/30, 1/40, 1/50, 1/60, 1/70, 1/80, 1/90, 1/100, 1/200, 1/300, 1/400, 1/500, 1/600, 1/700, 1/800, ...
11
votes
3answers
134 views

Doubt regarding divisibility of the expression: $1^{101}+2^{101} \cdot \cdot \cdot +2016^{101}$

In an interesting contest question I recently encountered, I chanced upon a question I couldn't solve. $$\sum^{2016}_{i=1}i^{101}$$ is divisible by: (a)2014 (b)2015 (c)2016 (d)2017 How would I ...
1
vote
0answers
12 views

Sum of Gamma functions (product pairs)

This is my first time asking a question on stackexchange. Is there an analytical expression for the following summation of Gamma functions? $\sum_{t=0}^m \Gamma (A + t) \Gamma (B + m -t) = ?$ for ...
1
vote
2answers
75 views

Find the sum of the series: $\frac{1}{1*2} - \frac{1}{3*2^3} + \frac{1}{5*2^5} - \frac{1}{7*2^7}+\dots$?

$$\frac{1}{1*2} - \frac{1}{3*2^3} + \frac{1}{5*2^5} - \frac{1}{7*2^7}+\dots$$ I made a series to get $$\sum_{n=0}^\inf \frac{(-1)^n}{(1+2n)*2^{1+2n}}$$ but what series can it manipulate and simplify ...
2
votes
6answers
75 views

Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent?

Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent? I tried ratio test but didn't seem to work, and I also know that the limit goes to zero, but I can't say its convergence because ...
-1
votes
0answers
10 views

Sigma on summation, how to find the equaling summationindeks

A given summationindeks and summationlimits (upper and lower) equal another given sigma with upper and lower limits. How to find the summationindeks of Right hand side?
0
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3answers
54 views

Evaluate $\frac{1-S_{2011}}{1-S_{2012}}$ when $S_n=\sum_{r=1}^{n}{\frac{r}{(r+1)!}}$

Suppose that, $$S_n=\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\ldots+\frac{n}{(n+1)!}$$ or more compactly, $S_n= \sum_{r=1}^{n}{\frac{r}{(r+1)!}}$. How can we find the value of ...
-2
votes
1answer
59 views

Value of sum of binomials: $P = \binom{N}{0}-\binom{N}{1}+\binom{N}{2}-\binom{N}{3}+ \dotsb + (-1)^N\binom{N}{N}$ [duplicate]

$P = \binom{N}{0}-\binom{N}{1}+\binom{N}{2}-\binom{N}{3}+ \dotsb + (-1)^N\binom{N}{N}$ I can calculate the value of this equation manually, but there any direct formula for calculating the value of ...
2
votes
2answers
50 views

Combinatorial identity for $\sum_{k = 0}^n \binom{n - k + a}{n - k} \binom{k + b}{k}$

I am interested in \begin{equation} \sum_{k = 0}^n \binom{n - k + a}{n - k} \binom{k + b}{k} \tag{1}, \end{equation} for non-negative integers $a$ and $b$. Mathematica tells me that ...
0
votes
2answers
58 views

Infinite Sum Calculation

How do we calculate this? $$ \sum_{n=1}^\infty \frac{n-1}{n!}$$ From my readings on previous posts here and through Google, I found that: $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ But then how do I ...
1
vote
1answer
28 views

Converting this summation into an integral

This summation includes a sum of n derivatives of the function f(x) at the point (c+d) / 2 I'm trying to convert a Taylor ...
2
votes
2answers
33 views

Proving this quotient is always even.

Let $n$ be an odd, composite integer. Let $D$ equals the sum of the proper divisors of $n$ minus the last divisor, $d_j$. Also, let $n$ be a number such that $D \gt \frac{1}{2} \times{n}$. Let $d_i$ ...
3
votes
4answers
98 views

Sum problem $S_n=1q+8q^2+27q^3+ … +n^3q^n$ [closed]

Can anyone help me with this sum problem. $$S_n=1q+8q^2+27q^3+ ... +n^3q^n$$ I tried to set this like geometric sequence but I couldn't do it. Without: Taylor
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vote
0answers
23 views

How does $J_{-n}(x)=(-1)^n J_{n}(x)$?

I am trying to understand why $J_{-n}(x)=(-1)^n J_{n}(x)$. In the following link http://oer.physics.manchester.ac.uk/PDEs/Notes/jsmath/Notesse37.html the author proves this by doing the following ...
2
votes
2answers
54 views

Summation $\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+…$

I came across a question today... Q. The sum $\dfrac{3}{1^2}+\dfrac{5}{1^2+2^2}+\dfrac{7}{1^2+2^2+3^2}+....$ upto $11$ terms is? Okay, I think it can be written as ...
1
vote
0answers
28 views

Is it possible to find the partial sum?

Let, $a_n=\frac {3^{n+1}}{1+2^{n+1}},n\geq0.$ Let $S_n$ be the partial sum defined by $$S_n=\sum_{i=0}^{n}a_i.$$ Is it possible to write a closed formula for $S_n.$ I have no idea how to do this. Any ...
0
votes
2answers
42 views

When is a finite sum of powers of non-integer a rational number? [closed]

Concretely, is there $ b \in \mathbb R, n,k \in \mathbb N $ such that $ \sum_{i = n}^{n+k} b^i \in \mathbb Q$ ?
0
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2answers
50 views

Could the sum of powers of non-integers result in a whole number? [closed]

Concretely, is there a $ b \in \mathbb R $ such that $ \sum_{i \in I \subset \mathbb N} b^i \in \mathbb W$ ?
0
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3answers
43 views

Proof by induction that $1^2 + 3^2 + 5^2 + … + (2n-1)^2 = \frac{n(2n-1)(2n+1))}{3}$

I need to know if I am doing this right. I have to prove that $1^2 + 3^2 + 5^2 + ... + (2n-1)^2 = \frac{n(2n-1)(2n+1))}{3}$ So first I did the base case which would be $1$. $1^2 = ...
0
votes
2answers
28 views

Why can't you do a breakdown method for division in summation notation?

$$\sum_{i=1}^5 i^2/5i$$ For this example, the answer is $3$ with simple plugging in and expansion. However, upon calculating ${n^2}$ and ${5n}$ separately and dividing I come up with: ...
1
vote
2answers
28 views

Probability of Rolling a $1$ on an $n$-Die in $r$ Attempts

I roll an $n$ sided die, numbered $1 \to n$. If I roll a $1$, I walk away; otherwise, I roll the die again. This process could repeat indefinitely. What is the probability, $P(n,r)$, that I will roll ...
4
votes
4answers
73 views

How to estimate $\sum_{n=1}^{\infty}\frac{(-1)^n}{n^3}$ with error less than $0.01$?

How to estimate $\sum_{n=1}^{\infty}\frac{(-1)^n}{n^3}$ with error less than $0.01$? In order to solve the question, I think we need to write out the terms. So ...
1
vote
1answer
39 views

Alternating sign odd number generating function.

I have a sequence that I'm trying to find both an ordinary generating function for as well as a closed form without a floor function. The sequence $$\{1,1,-1,3,-3,5,-5,7,-7,9,-9,11,-11,\}$$ is ...
0
votes
3answers
54 views

Finding a sum to infinity with a factorial

The sequence is $$\sum_{n=1}^{\infty} \frac{n-1}{n!}$$ I could show that it converges using the Ratio test, but evaluating it seems to be hard. I'm trying to avoid power series and Taylor series ...
0
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0answers
16 views

Summation with a variable as upper limit

I have done summations without a variable as the upper limit, but a question like this is an example problem that I can't figure out. What steps do I go through to solve this? $$\sum_{j=1}^m j(2j+3) ...
0
votes
0answers
28 views

Can we derive a summation counterpart of an integral in general?

Is there any general or heuristic rule to convert a function or equation involved with integrals to another one with summations? How can we deal with the case when the Jacobian matrix show up? I am ...
1
vote
1answer
41 views

What is the relation between the square root of the sum of squares and the sum of the absolute values?

I want to prove that $\sqrt{\sum a_{i}^{2}} \geq \sum \left | a_{i} \right |$, is it possible ?
9
votes
3answers
888 views

How do I manipulate the sum of all natural numbers to make it converge to an arbitrary number?

I just found out that the Riemann Series Theorem lets us do the following: $$\sum_{i=1}^\infty{i}=-\frac{1}{12}$$But it also says (at least according to the wikipedia page on the subject) that a ...
4
votes
4answers
107 views

Can anyone give a combinatorial proof of the identity ${n \choose m} + 2{n-1 \choose m}+3{n-2 \choose m}+…+(n-m+1){m \choose m}={n+2 \choose m+2}$

Can anyone give a combinatorial proof of the identity $${n \choose m} + 2{n-1 \choose m}+3{n-2 \choose m}+\ldots+(n+1-m){m \choose m}={n+2 \choose m+2}$$ I am finding difficult as $n$ is varying ...
0
votes
0answers
15 views

How many items can I put into 5076 boxes of size $x_i = x_{i-1} +\frac{x_{i-1}}{1024}$?

I have a virtual 'block' where I can put $x_0=225$ items into it. I have a chain of 5076 blocks, one after another. The number of items I can put into each block increases by $x_i = x_{i-1} ...
0
votes
1answer
74 views

Integral representation $f(x) = \lim_{n = \infty} \sum_1^n \ln(n)^2 + n \ln(x)^2 - \sum_1^n \ln(x+n)^2$?

Let $x>1$ be a real variable. $$f(x) = \lim_{n = \infty} \sum_1^n \ln(n)^2 + n \ln(x)^2 - \sum_1^n \ln(x+n)^2$$ Is there an integral representation for $f(x)$ that uses no $\sum$ , named ...
1
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0answers
19 views

Solving function which results singularity during discrete summation

I am sorry if the following query is too basic or technically incorrect. Actually I am quite away from mathematics so I could not figure out how to solve this. I have the following function. ...
0
votes
1answer
17 views

Numerator of Sample Variance Expectation

Suppose $w_1, \dots, w_k$ are iid with mean $\mu_w$, variance $\sigma^2_w$. I understand the proof that $$\mathbb{E}\left[\sum_{i=1}^{k}(w_i-\bar{w}_{\cdot})^2\right] = (k-1)\sigma^2_w$$ where ...
3
votes
2answers
102 views

sum of series $\sum \limits_{i=1}^{n}\frac{i(i+1)}{2}$ [closed]

Does there exist an explicit formula for the sum of the series $$\sum \limits_{i=1}^{n}\frac{i(i+1)}{2}$$
0
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1answer
29 views

Resource management

Basically a vector where start with a capacity for B objects. When the vector runs out of space we reallocate with enough space for ...
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5answers
90 views

if $\sum\limits_{i=1}^n{x_i} = 1$, how do you choose the $x_i$'s such that $\sum\limits_{i=1}^n{x_i^2}$ is minimized?

if $\sum\limits_{i=1}^n{x_i} = 1$, how do you choose the $x_i$'s such that $\sum\limits_{i=1}^n{x_i^2}$ is minimized. I have an intuition that each $x_i = \frac{1}{n}$, but I don't know how to prove ...
3
votes
5answers
79 views

Tricky Nested Summation

I am trying to analyze a programming algorithm that has three nested loops. The algorithm looks like this: $$\sum_{i=0}^{n} \sum_{j=i}^{n}\sum_{k=i}^{j} 1$$. I am trying to simplify it. These are the ...
0
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0answers
31 views

Sums of fractions with $(-1)^{k}$ and products in denominators

How to calculate these sums? $$ \sum_{k=1}^{n}\frac{(-1)^{n+k}\cdot 2^{k(k-1)}}{\Bigg((2^2-1)(2^4-1)\ldots (2^{2k-2}-1)\Bigg)\cdot \Bigg((2^2-1)(2^4-1)\ldots (2^{2n-2k}-1)\Bigg)}; $$ $$ ...
4
votes
1answer
46 views

Prove that this summation evaluates out to $\zeta(2)-1$

I am aware of the following identity: $$\sum_{m=1}^\infty \left(\frac{1}{m}-\left(\zeta(2)-\sum_{n=1}^m \frac{1}{n^2}\right)\right)=\zeta(2)-1$$ I can't quite figure out how to prove this result. ...
2
votes
2answers
156 views

proof that the binomial sum is equal to 1

I'm trying to prove the following identity: let $k$ and $s$ be positive integers and let $k\ge s\ge 1$ $$\sum_{i=0}^{k-s} (-1)^i{s-1+i \choose s-1}{k \choose s+i} = 1$$ I've tried to use a ...
1
vote
4answers
36 views

Help with Summation Inequality: $\sum a_i \sum b_i \ge \sum a_ib_i$

I can't find a reference to the fairly simple idea that: $\Sigma (a_i) \Sigma(b_i) \geq \Sigma(a_ib_i)$ for $a_i,b_i \geq 0$ This is obviously equal for the case where $i$ goes only to 1 and for the ...
1
vote
1answer
42 views

Triples of natural numbers with same sum and product

Im looking at pairs of triples of natural numbers without repititions such that the sums of the two triples are equal and the products of the two triples are equal. To be precise: Let $x<y<z$ ...
1
vote
2answers
46 views

sum of the series $\frac{2^n+3^n}{6^n}$ from $n=1$ to $\infty$

Find the sum of the series $\sum_{n=1}^{\infty} \frac{2^n+3^n}{6^n}=?$ My thoughts: find $\sum_{n=1}^{\infty} 2^n$, $\sum_{n=1}^{\infty} 3^n$ and $\sum_{n=1}^{\infty} 6^n$ (although I don't know how ...
3
votes
2answers
277 views

Is knowing the Sum and Product of k different natural numbers enough to find them?

Can we uniquely identify the set of k different natural numbers (no two are the same) by knowing only their sum and product (and the value of k itself)?
1
vote
2answers
29 views

Take the derivative of a Hamming Weight Enumerator

Background: The Hamming weight enumerator can be written as such: $$A(z) = A_0 + A_1z + A_2z^2 + ... + A_nz^n$$ With $A_i$ being equal to the number of code words of weight i in the code book for an ...
0
votes
0answers
25 views

Computer Vision Models 4.7 - Simplification of Summations

I am reading through the Computer Vision: Models, Learning, and Inference book written by Simon J.D. Prince to get an understanding of computer vision. The author gives some examples in deriving the ...
1
vote
0answers
31 views

Integer solutions of $\sum_{j=1}^m\frac{n-m}{j}+\sum_{j=m+1}^n\frac{n}{j}=k$

This is in reference to an answer I gave to this question. I am curious to know if my intuition is correct. Given $n,m\in\mathbb{N}$, with $n>m$, is it possible for $\exp(2\pi ...
0
votes
1answer
42 views

Resemblance between $\sum\limits_{n=1}^\infty \sin(2nx)$ and $\cot x$

Is there a reason why this sum $$\sum\limits_{n=1}^\infty \sin(2nx)$$ Has some resemblence to the cotangent function?
2
votes
1answer
64 views

Calculation of sum with many indexes

How to calculated this sum in the closed form? $$ \sum_{(i_1,i_2,\ldots,i_k)\atop 1\leq i_1<i_2<\ldots<i_k\leq n} 2^{2(i_1+i_2+\ldots+i_k)-k} $$ Here $n$ is positive integer, $1\leq k\leq ...
1
vote
3answers
85 views

Using the Principle of Mathematical Induction to Prove propositions

I have three questions regarding using the Principle of Mathematical Induction: Let $P(n)$ be the following proposition: $f(n) = f(n-1) + 1$ for all $n ≥ 1$, where $f(n)$ is the number of subsets ...