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

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2answers
66 views

Summation of an infinite series

The sum is as follows: $$ \sum_{n=1}^{\infty} n \left ( \frac{1}{6}\right ) \left ( \frac{5}{6} \right )^{n-1}\\ $$ This is how I started: $$ = \frac{1}{6}\sum_{n=1}^{\infty} n \left ( \frac{5}{6} ...
3
votes
1answer
39 views

$\sum_{i=0}^{k} \binom{m}{i}\binom{n}{k-i} =\binom{m+n}{k}$ [duplicate]

I'm trying to show that the equality $$\sum_{i=0}^{k} \binom{m}{i}\binom{n}{k-i} =\binom{m+n}{k}$$ Is true. I know it is since there is a good combinatorical argument for it. If we have a group of ...
1
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0answers
37 views

Partial sum formula for $\sum_{n=0}^{x} {\tan(x)}$ from $(0,\infty)$

I know there is no elementary way of expressing the partial sums of $\tan(x)$. I know; however, I can get an approximation of partial sums using a series, such as the MacLaurin series. If a series can ...
3
votes
4answers
101 views

Calculate $ S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}. $

Calculate $S =\displaystyle\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}$. This sequence is neither arithmetic nor geometric. How can you solve this. Thanks!
2
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0answers
16 views

How to find a $\theta$ function verifying this property?

Let $r>4$ and $n>1$ be positive integers. Intuitively, the infinite sum $$S=\sum_{m=1}^{∞}\frac{2m}{r^{m^2}}$$ is related to a $\theta$ function. However, I cannot find a way to calculate this ...
2
votes
1answer
52 views

Can we simplify this sum?

Let $r>4$ and $n>1$ be positive integers. Can we simplify this sum: $$S=\sum_{m=1}^{n}\frac{2m}{r^{m^2}}$$ I have no idea to start.
3
votes
3answers
65 views

Showing ${n + a - 1 \choose a - 1} = \sum_{k = 0}^{\left\lfloor n/2 \right\rfloor} {a \choose n-2k}{k+a-1 \choose a-1}$

Prove that for integers $n \geq 0$ and $a \geq 1$, $${n + a - 1 \choose a - 1} = \sum_{k = 0}^{\left\lfloor n/2 \right\rfloor} {a \choose n-2k}{k+a-1 \choose a-1}.$$ I figured I'd post this question, ...
1
vote
1answer
26 views

Evaluation of an Infinite Converging Sum

I was asked to find the value of the following summation. $$ x=1+\sum_{i=1}^{\infty}\frac{1}{2^i}+\sum_{j=1}^{\infty}\frac{1}{3^j}+\sum_{k=1}^{\infty}\frac{1}{5^k} $$ I "solved" it approximately by ...
4
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3answers
45 views

Hypergeometric 2F1 with negative c

I've got this hypergeometric series $_2F_1 \left[ \begin{array}{ll} a &-n \\ -a-n+1 & \end{array} ; 1\right]$ where $a,n>0$ and $a,n\in \mathbb{N}$ The problem is that $-a-n+1$ is ...
3
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0answers
35 views

Finite sum of floor function. [duplicate]

Suppose I have the following finite sum; $$S_n=\sum_{k=1}^n{\left\lfloor{\frac{n}{k}}\right\rfloor}$$ I've never dealt with something like this before and was curious of a way to express it with a ...
1
vote
2answers
44 views

Compute the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$

What would be the sum of the following series? $$\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$$ Thanks
3
votes
1answer
58 views

Induction proof about entries of powers of strictly upper triangular matrix

Let $A$ be a $n \times n$ strictly upper triangular matrix. Prove that, for $k \ge1$, the matrix $A^k$ has the property that $(A^k)_{i,j} = 0$ for all $(i,j)$ with $j-i < k$. Also, show that $A^n ...
2
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4answers
66 views

Simplify $f(x) = \sum_{n=0}^{\infty}{\frac{\sin{(nx)}}{n!}}$

Can somebody give me a hint about what technique I need to simplify this summation? Thanks
1
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2answers
57 views

Asymptotic of $\sum_{n = 1}^{N}\frac{1}{n}$ versus $\sum_{1 \leq n \leq x}\frac{1}{n}$

Consider the series $\sum_{n = 1}^{N}\frac{1}{n}$. It is well known that we have the asymptotic: $$\sum_{n = 1}^{N}\frac{1}{n} = \log N + \gamma + \frac{1}{2N} + \frac{1}{12N^{2}} + O(N^{-3}).$$ My ...
0
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1answer
27 views

How are sinusoids and roots of unity related to each other?

The discrete Fourier transform (DFT) is often teached as being a transform that decomposes a given signal or sequence of numbers into sinusoids with frequencies $\large\frac{k}{N}$ where $k \in [0, ...
0
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3answers
76 views

Showing $1 + 4\cos{2\theta} + 6 \cos{4\theta} +4\cos{6\theta}+ \cos {8\theta} =16\cos{4\theta} \cos^4 \theta$.

Show that $1 + 4\cos{2\theta} + 6 \cos{4\theta} +4\cos{6\theta}+ \cos {8\theta} = 16\cos{4\theta} \cos^4 \theta$ Do I solve this question by using summation of series?
0
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1answer
65 views

Given that $\sum\limits_{i=1}^{n}x_i=m+r$, show that $\sum\limits_{i=1}^{n}x_i^2\leq{m+r^2}$

The summation of real numbers $x_i\in (0,1)\, \text{for}\, i=1,\ldots ,n$ is equal to $m+r$, where $m$ is an integer and $r\in [0,1)$. Show that $$\sum_{i=1} ^n x_i^2\leq m+r^2.$$ I pick up this ...
0
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1answer
31 views

Calculating Area using Sigma Notation

This is my second time posting. Thanks again for answering my first question. This is my question: Find the area above y = 0, from x = -1 to x = 1 of y = 2^x. Use subintervals of equal length. I ...
2
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3answers
63 views

Proof By Induction of a Sum

Can someone look at my proof. I am supposed to prove by induction. The question is to prove the following: $$\sum _{i=0}^{n}{i} =\frac { n\left( n+1 \right) }{ 2 } .$$ If $n=1$ Then ...
2
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3answers
53 views

Evaluate: $\sum_{k=2}^n\frac {n!}{(n-k)!(k-2)!}$

Evaluate: $$\sum_{k=2}^n\frac {n!}{(n-k)!(k-2)!}$$ Attempt $S_2=\frac {n!}{(n-2)!}$ $S_3=\frac {n!}{(n-3)!}$ $S_4=\frac {n!}{2(n-4)!}$ $\vdots$ $S_{n-1}=\frac {n!}{1!(n-3)!}$ $S_n=\frac ...
2
votes
2answers
62 views

Prove that $\sum_{k=1}^{m}\frac{1}{k(k+1)}=1-\frac{1}{m+1}$.

I know $\sum_{k=1}^{m}\frac{1}{k(k+1)}=1-\frac{1}{m+1}= H_mH_{m+1}$, for $H_m$ the harmonic sum. I tried many ways to prove it like this $$\sum_{k=1}^{m}\left(\frac{1}{k(k+1)} + ...
1
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1answer
15 views

Sigma notation abstract limits

What is meant by the following notations? $\sum\limits_{k=0}^{p+1}k^3$, $\sum\limits_{k=0}^{p-2}k^3$ I need to use this notation to prove a statement is true for $\sum\limits_{k=0}^{p+1}k^3$ when ...
1
vote
1answer
43 views

Why is $\sum_{m=0}^k \frac{k^2+k}{2} = \frac{(k^2+k)(k+1)}{2}$?

I recieved an answer like this : $S= \displaystyle \sum_{m=0}^k \dfrac{k^2+k}{2} - k\displaystyle \sum_{m=0}^k m + \displaystyle \sum_{m=0}^k m^2=\dfrac{(k^2+k)(k+1)}{2}-\dfrac{k\cdot ...
0
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5answers
58 views

Why isn't $\sum_{m=0} ^{k}\frac{k^2 + k -2mk +2m^2} {2}=\frac{(-3k+1)(2k^2+1)}{12}?$

The right equation is $$\sum_{m=0} ^{k}\frac{k^2 + k -2mk +2m^2}{2}=\frac{k(k+1)(k+2)}{3}?$$ In my calculation $$\sum_{m=0} ^{k}\frac{k^2 + k -2mk +2m^2}{2}=\frac{(-3k+1)(2k^2+1)}{12}.$$ I don't ...
0
votes
4answers
56 views

Why is $ \sum_{n=0}^{k}|m-n|=\sum_{n=0}^{m}(m-n)+\sum_{n=m}^{k}(n-m)$? [closed]

The problem is: Why is $$ \sum_{n=0}^{k}|m-n|=\sum_{n=0}^{m}(m-n)+\sum_{n=m}^{k}(n-m)\;?$$
0
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1answer
17 views

Calculate variance for effort estimation

I want to calculate the variance for effort estimation (Scrum), but unfortunately I get a wrong result. $$ \text{Variance} = \displaystyle \frac{1}{n} \sum_{i=1}^n (X_i - ev)^2 $$ Please note: $n$: ...
0
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1answer
45 views

Algebra help on Summation

The following is my professor's time complexity for insertion sort. I need help on the summation part. Please can someone help me understand the distribution of the $c_5$ and $c_6$ constants ...
0
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3answers
28 views

Summation/Sigma Notation Question

This is my first time on here. I am in first year engineering, and I'm having some trouble with sigma notation. Here is the question and answer: I am trying to convert the summation into closed ...
1
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3answers
68 views

Evaluate to find the sum of an infinite series [duplicate]

$∑_{n=1}^\infty$ $n\over2^{n-1}$ or 1 + $2\over2$ + $3\over4$ + $4\over8$ + $5\over16$ + $\ldots$ How to go about evaluating the above, showing that it sums to 4?
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4answers
40 views

Prove Summation to Some Number $n$ [duplicate]

Let $n\in \mathbb{N}$. Can someone help me prove this by induction: $$\sum _{i=0}^{n}{i} =\frac { n\left( n+1 \right) }{ 2 } .$$
11
votes
3answers
236 views

On the sum of digits of $n^k$

Reading another question on the sum of the digits of $2^n$ i started wondering wether there exist a $\alpha\in\mathbb{N}$ such that for every $n>\alpha$ we have $S(2^{n+1})>S(2^n)$, where $S(n)$ ...
0
votes
0answers
35 views

Summation with Binomial Coefficients, $\sum (-1)^k \binom{m_1}{k} \binom{m_2}{k} $

I have trouble doing this summation: $$ \sum_{k=0}^{\min(m_1,m_2)} (-1)^k \binom{m_1}{k} \binom{m_2}{k} $$ where $m_1$ and $m_2$ are positive integers. Can someone help?
0
votes
0answers
9 views

Generalized Hurwitz Zeta function

Let $d\ge 1$ be an integer, $a>0$ be a real number and let $\vec{s} := (s_0,\cdots,s_{d-1})$ where all the components are strictly bigger than one. We generalize the zeta function to higher ...
0
votes
1answer
27 views

How to prove $ \sum_{k=1}^{p+1} \binom{p+1}{k}S_{n}^{p+1-k} = (n+1)^{p+1}-1$?

This is the solution $ \sum_{k=1}^{p+1} \binom{p+1}{k}S_{n}^{p+1-k} = \sum_{k=1}^{p+1} \binom{p+1}{k} \sum_{l=1}^{n}l^{p+1-k} = \sum_{l=1}^{n}(l+1)^{p+1}-l^{p+1} = (n+1)^{p+1}-1$ ? With $S_{n}^{p} = ...
2
votes
4answers
134 views

How to find the value of this summation equation?

The question is: $$\sum_{i=1}^n (i^2+3i+4)$$ I get that $$\sum_{i=1}^n i^2 = \frac{n(n+1)(n+2)}{6}$$ and $$3\sum_{i=1}^n i = \frac{3n(n+1)}{2}$$ so one would get I'll call this form1: ...
6
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2answers
210 views

Using the rules that prove the sum of all natural numbers is $-\frac{1}{12}$, how can you prove that the harmonic series diverges?

I think I understand intuitively how we can assign a value to the sum of all natural numbers. But of all the proofs that I've seen that show why $\zeta(-1) = -\frac{1}{12}$, none of them use their own ...
1
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2answers
92 views

Is $\bigl(\sum {{x^n}\over{n!}} \bigr) \bigl(\sum {{y^n}\over{n!}} \bigr) = \bigl(\sum {{(x+y)^n}\over{n!}}\bigr)$ generalizable for series?

Before I had to do a proof demonstrating the properties of exponential multiplication using power series expansions: $$ e^xe^y=e^{x+y}, $$ and the easiest and quickest way I could think of doing this ...
0
votes
1answer
25 views

Closed form for this partial polynomial sum? $\sum_{i=0}^{k-1} {n + i - 1 \choose i} x^i$

I came across a sum: $$p_k(x, n) = \sum_{i=0}^{k-1} {n + i - 1 \choose i} x^i$$ and I was wondering if it had a closed form. I found on wikipedia: $$\sum_{i=0}^{\infty} {n + i - 1 \choose i} x^i = ...
1
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2answers
72 views

Question about Binomial Sums [duplicate]

Prove that for any $a \in \mathbb{R}$ $$\sum_{k=0}^n (-1)^{k}\binom{n}{k}(a-k)^{n}=n!$$ I rewrote the sum as $$\sum_{k=0}^n \left((-1)^{k}\binom{n}{k} \sum_{i=0}^n (-1)^{i}a^{n-i} k^{i} ...
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votes
1answer
53 views

What is $\sum_{n=0}^\infty (-1)^n$ [closed]

$\sum_{n=0}^\infty (-1)^n$??? Give the answer with method also
-1
votes
4answers
79 views

Deriving formula for sum n(n+1) [duplicate]

Can you please describe how to derive a formula for first n members of $$ S = 1\cdot 2 + 2\cdot 3 + 3\cdot 4 +\cdots +n(n+1)\mbox{?} $$ Thank you
0
votes
1answer
44 views

(0 - Sum) Representable by Numbers in a Set

Given: K is a finite set of positive integers. The sum of all numbers in the set can be found, S. Given: $K_i$ is the subset of K containing all numbers less than $2^i$. The sum of all numbers in ...
1
vote
2answers
102 views

How to calculate $\sum\limits_{n=-\infty}^\infty \frac{1}{1+n^2}$ [closed]

How can I calculate the summation: $\displaystyle \sum\limits_{n=-\infty}^\infty \frac{1}{1+n^2}$
1
vote
1answer
24 views

Prove by induction that $\sum_{\varnothing\ne S\subseteq[n]}(\prod S)^{-1}=n$.

I'm having a hard time visualizing how to prove the following by induction: For every positive integer $n$, let $[n]$ denote the set $\{1,\ldots,n\}$. Let $A$ be a set. Use the notation $P(A)$ ...
4
votes
2answers
232 views

Nasty Limit of sum at infinity

$$\lim_{n \to \infty} \left (\sum_{i=1}^n \frac{a\left[\left(\frac{b}{a}\right)^{\frac{i}{n}}-\left(\frac{b}{a}\right)^{\frac{i-1}{n}}\right]}{a\left(\frac{b}{a}\right)^{\frac{i-1}{n}}}\right) $$ How ...
1
vote
3answers
79 views

Finite summation with binomial coefficients, $\sum (-1)^k\binom{r}{k} \binom{k/2}{q}$

I came across the following finite sum involving (generalized) binomial coefficients: $$ 2^q \sum_{k=0}^r \binom{r}{k} \binom{k/2}{q} (-1)^k .$$ Putting this into Mathematica gives me: $$ (-1)^q ...
18
votes
1answer
378 views

Fibonacci-related sum

Related to this question Find a solution for f(1/x)+f(1+x)=x, what is this sum: $$\sum_{n=1}^{\infty}(-1)^n\left(\frac{F_n}{F_{n+1}}-\frac1{\phi}\right)$$ where $F_n$ is the $n$th Fibonacci number and ...
5
votes
0answers
45 views

A summation involving the ceiling function

I'm trying to find a better method of calculating the sum $$\sum_{k=1}^N\lceil ak\rceil^2$$ where $a$ is an irrational number. So far, my only idea is to somehow use a best rational approximation. ...
1
vote
1answer
47 views

Summation to Equation

I have a summation and I want to be able to find the sum for given $n$ without having to go through $1,\dots,n$. $$\sum_{x=1}^{n - 1}x+300\cdot2^{x/7}$$ It's been awhile since I've done summations ...
2
votes
3answers
60 views

Is there any way to get convergent series if I know the sum?

If I am given the limit of a convergent series is there any way that I can find the series? Is it possible that for any given limit there are infinitely many or no solutions at all?