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
28 views

Summation from $2^0 log_2(n)$ to $2^{n-1} log_2(1)$ [on hold]

What is the sum: $2^0 log_2(n) + 2^1log_2(n-1) + 2^2log_2(n-2) + ... + 2^{n-2}log_2(2) + 2^{n-1}log_2(1)$?
-5
votes
7answers
47 views

What is $1+2+4+8+16+…+2^n$? [duplicate]

What is the result of: summation from one, two, four, eight until $n$ power of two? Thank you!
1
vote
2answers
42 views

How can I solve $\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$ step by step?

$$\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$$ First off I know the answer to this question, but I do not know how to actually get the answer I do know some basic sums like that one. Can someone ...
1
vote
1answer
36 views

If $(a+b)^n=\sum_{k=0}^{n}{n\choose k}a^{n-k}b^kc_k$, then $c_k=1$?

Be advised this is a real soft question: If $$(a+b)^n=\sum_{k=0}^{n}{n\choose k}a^{n-k}b^kc_k$$ Assuming $abc \neq 0$ must we have the following condition? $$c_k=1$$ for all $0 \leq k \leq n$ How do ...
1
vote
4answers
35 views

Solving this Recurrence Relation in terms of previous values.

What will be the value of $X(n)$ and $Y(n)$ in terms of given $n,X(0),Y(0)$. $$ X(n) = X(n-1) + Y(n-1) \\ Y(n) = 2X(n-1) + Y(n-1) $$
3
votes
2answers
51 views

Sum involving zeta functions

Find closed form of the following - $$ \displaystyle \sum_{n=2}^{\infty}{\left(\frac{(n-1)\zeta(n)}{4n-1}\right)} $$ I don't know how to approach to it - Using the integral definition? I cannot use ...
0
votes
2answers
26 views

Simplifying with Summation

This is a problem out of my statistics book but my issue is simplifying from Step 3 to Step 4 below: Step 1: var X=$\sum\:p_i\:(x_i-E[X])^2$ Step 2: var X=$\sum\:p_i[x_i^2+E[X]^2-2x_iE[X]]$ Step ...
3
votes
1answer
36 views

bounding a sum using a definite integral

Conjecture. Let $1<p<\infty$. Then there exists $C\in(0,\infty)$ such that for any $k\in\mathbb{Z}^+$ we have \begin{equation}\sum_{n=1}^k(k+1-n)^{-\frac{p}{p+1}}n^{-\frac{1}{p+1}}\leq ...
5
votes
2answers
150 views

What is the integer part of $\sum_{i=2}^{9999} \frac {1}{\sqrt i}?$

What is the integer part of: $$\sum_{i=2}^{9999} \frac {1}{\sqrt {i}}$$ A short but tricky problem. Any help is welcome.
0
votes
0answers
18 views

Does this theorem concerning upper and lower bnound of a monotone decreasing function have a formal name?

This is the theorem: Let $g$ be a monotone decreasing function and let $a,b \in \mathbb{N}$. Then the following holds true: $$\int_{a}^{b+1}g(x)dx \overset{(i)}{\leq} ...
3
votes
3answers
50 views

Does $\lim\limits_{n \to +∞} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$ diverge?

Does the limit of this summation diverge? $$\lim\limits_{n \to +∞} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$$ Thanks!
1
vote
2answers
32 views

Question about summations with an unknown lower limit

I am unsure of how to proceed about finding the solution to this problem. $$\sum_{i=6}^8(\sum_{j=i}^8 (j+1)^2)$$ Obviously the last step is not to difficult, but the fact that the lower limit for the ...
1
vote
2answers
47 views

Solve $\frac{1}{2^\theta}\sum_{k=0}^{\theta} {\theta\choose k} \delta(k)=\theta$ for $\delta$

The following arises in unbiased estimation of a parameter for the binomial distribution, but that information is not needed for solving the question. I tried solving this by manipulating the sum to ...
2
votes
4answers
123 views

Prove that $1.49<\sum_{k=1}^{99}\frac{1}{k^2}<1.99$

It can be proven by induction that $$\sum_{k=1}^{n}\frac{1}{k^2}\leq2-\frac{1}{n}$$ From here, we can easily acquire the upper bound of the sum $$\sum_{k=1}^{99}\frac{1}{k^2}$$ letting $n=100$. ...
0
votes
1answer
27 views

Combined pulling power of multiple engines in a train

Apologies if this question doesn't belong here.. Its very common to see multiple engines being used when there are more coaches in a train. In a configuration with more than one engine a good amount ...
0
votes
1answer
15 views

How many ways are there to express a natural as a sum of 3 others—but by induction?

I have figured out an (inductive?) process, but I cannot express it formally: There is always one possibility where $n$ is in the first place of our 3-tuple: $[n~~0~~0]$. Then I can subtract $k~(\leq ...
-3
votes
1answer
33 views

What is the value of this sum? $\sum_{r=k}^{\infty} \frac{1}{r^{3/2} e^{\frac{c k}{2 r}}}$ [on hold]

How can I calculate this, where $c>0$? $$ \sum_{r=k}^{\infty} \frac{e^{\frac{-c k}{2 r}}}{r^{3/2}} $$
3
votes
2answers
62 views

How is the Radius of Convergence of a Series determined?

Consider $$\sum_{n=0}^{\infty}\frac{(-1)^nx^n}{(n+1)^2}$$ which by the ratio test the ratio of two consecutive terms converges to $|x|$ as $n\rightarrow \infty$ and has a radius of convergence equal ...
2
votes
4answers
134 views

Summing n times binomial(n,k)

I'm trying to do $\sum_{n=a}^b \left( \begin{array}{rl} n \\ a \end{array} \right) n $ . Is there a formula, that anybody knows?
0
votes
0answers
45 views

Binomial Sum Formula

I can't find a good closed form expression for this, $\sum_{k=0}^n\left[\binom{n}{k}\binom{m}{k}\right]$, where n is the variable, and m is a fixed constant, to be included in the formula. :( Can ...
4
votes
4answers
98 views

Combinatorial Proof for Binomial Identity: $\sum_{k = 0}^n \binom{k}{p} = \binom{n+1}{p+1}$ [duplicate]

I am studying combinatorics and I came across the identity $$\sum\limits_{k=0}^n \binom kp =\binom {n+1}{p+1}.$$ I have read the algebraic proof and it does not appeal to me. Is there an elegant ...
4
votes
0answers
60 views

On finding an explicit form of a particular recurrence relation

Let $f$ be integrable over the interval $[0, 1]$, and $$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$ Suppose $f(x) = f(1-x)$; we can then show that $$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, ...
1
vote
3answers
73 views

Surprising Summation (3): $\sum_{i=1}^n\sum_{j=1}^i 2(n-i)+1=\sum_{i=1}^n i^2$

Show, without expanding the summation, that $$\sum_{i=1}^n\sum_{j=1}^i 2(n-i)+1=\sum_{i=1}^ni^2$$ It would be interesting to see different approaches to this problem, when expansion is not ...
6
votes
1answer
124 views

Does $(1^a+2^a+3^a+4^a+5^a)^b=1^c+2^c+3^c+4^c+5^c$ imply $(a,b,c)=(1,2,3)$?

Question : Is the following proposition true? Proposition : For positive integers $a,b,c$ where $b\ge 2$, if $$(1^a+2^a+3^a+4^a+5^a)^b=1^c+2^c+3^c+4^c+5^c$$then $(a,b,c)=(1,2,3)$. This is ...
1
vote
3answers
72 views

Missing steps: Show the sum of the first n positive integers is of order $n^2$

In Rowsen's Discrete Mathematics text, 6th edition. He has this problem as an example (#11) on page 190. His solution for obtaining a lower bound is to ignore the first half of the terms. He does the ...
0
votes
2answers
48 views

find the coefficient

If $n$ is an odd natural number, and $\sin(n\theta) = \Sigma_{r=0}^{n} b_r \sin^r\theta$, then find $b_r$ in terms of $n$. I have tried this using trigonometric expansion but unable to find solution ...
2
votes
2answers
49 views

Is $S(a \cdot b)$ always less than $S(a) \cdot S(b)$

Let $S$ be the sum of digits of a natural number. I'm wondering if we have $$\forall (a,b) \in \mathbb{N}^2,S(a\cdot b)\le S(a)\cdot S(b)$$ I tried with some examples and for the moment it works, ...
1
vote
2answers
85 views

Solving $x^{2n} = \frac{1}{2^n}$

What is the principle behind solving for a variable that is raised to another variable? I came across this problem doing infinite sums: I had to solve the equation $$x^{2n} = \frac{1}{2^n}$$ for ...
1
vote
3answers
198 views

How could I find the sum of this infinite series by hand?

$$\sum_{n=1}^{\infty}\frac{(7n+32)3^n}{n(n+2)4^n}$$ Thank you!
2
votes
0answers
33 views

Please help me understand Analytic Density $\lim_{\sigma \to 1^+}\frac{1}{\zeta(\sigma)}\sum_{n \in A} \frac{1}{n^{\sigma}}$

$d (A) = \lim_{\sigma \to 1^+}\frac{1}{\zeta(\sigma)}\sum_{n \in B} \frac{1}{n^{\sigma}}$ for $B \subset \Bbb{N}$. So clearly this limit is $0$ for reciprocally summable (convergent) $B$. My goal ...
2
votes
0answers
35 views

How to solve this equation with implicit sum

I want to know how the authors of this arxiv paper (p. 10) solved the equation \begin{align} g\left(\lambda\right) ={}& ...
3
votes
3answers
93 views

Upper and lower bounds for $S(n) = \sum_{i=1}^{2^{n}-1} \frac{1}{i} = 1+\frac{1}{2}+ \cdots +\frac{1}{2^n-1}.$ [duplicate]

For a positive integer $n$ let $S(n) = \sum_{i=1}^{2^{n}-1} \frac 1i = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+ \cdots +\frac{1}{2^n-1}.$ Then which of the following are true. (a) ...
0
votes
1answer
39 views

How do I show this function is monotonically decreasing?

Let $p = $Probability of head on a coin toss ; $p < 0.5$ (biased coin). $f(k) =$ Probability that heads is the majority from $k$ tosses, where $k$ takes odd values. I want to show that $f(k)$ ...
1
vote
2answers
80 views

regularization of sum $n \ln(n)$

I was testing out a few summation using my previous descriped methodes when i found an error in my reasoning. I'm really hoping someone could help me out. The function which i was evaluating was ...
0
votes
1answer
27 views

Weighted sum of angles modulo $\pi/2$

Angle modulo $\pi /2$ means: $(a+ \pi /2) \mathbin{\%} \pi/2=a$, $a \in [0, \pi/2)$, which could be illustrated as a ‘modulo circle’ in the following figure. How to calculate the weighted sum of a ...
1
vote
2answers
78 views

Prove the Identity $\pi=2- \sum_{1}^{∞} \frac{(-1)^m}{m^2-\frac{1}{4}} $

By considering the fact that $f(\pi/2)=1$, prove the identity $\pi=2- \sum_{1}^{∞} \frac{(-1)^m}{m^2-\frac{1}{4}} $ This question was is a subsection in a chapter on Fourier series, can I use my ...
3
votes
2answers
52 views

Sum over two binomials identity

So while trying to count the number of configurations in a statistical mechanics research problem I come across this lovely sum: $$\sum_{i=0}^k \binom{i+r}{r} \binom{k-i+r}{r}$$ I scoured the ...
0
votes
1answer
18 views

Change of indices in a double summation

By using a "smart" change of indices $i$ and $j$, I'm trying to show that \begin{equation} \sum_{i=1}^{N}\sum_{j=1}^{N}q_{i}q_{j}a_{i}\left(f_{i}f_{j}^{'}-f_{i}^{'}f_{j}\right) = ...
2
votes
2answers
39 views

Binomial sum with two parameters

Let $m$ and $n$ be two integers. Evaluate $$S_{m,n}=\sum_{j=0}^{m} (-1)^j \binom{m}{j}\binom{mn-jn}{m+1}$$ At first, for $n=2$ I got $S_{m,2}=2^{m-1}m$, for $n=3$ I obtained $S_{m,3}=3^m m$, then I ...
13
votes
3answers
226 views

What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?

Find a closed form expression for $$\sum_{r=0}^n \dfrac{(-1)^r}{\dbinom{n}{r}}$$ where $n$ is an even positive integer. I tried using binomial identities, but since the binomial ...
2
votes
1answer
26 views

Prove or disprove: $ \sum_{b \vee d = x} \tau(b) \tau(d) = \tau(x)^3$

Can somebody prove or disprove? Let $\tau$ be the divisors function, so that $\tau(6) = \#\{ 1,2,3,6\} = 4$ $$ \sum_{b \vee d = x} \tau(b) \tau(d) = \tau(x)^3$$ Here I am using $b \vee d = ...
0
votes
2answers
56 views

How I can simplify this double sum?

How I can simplify this double sum: $$S=\sum_{k=1}^{n-1}\sum_{l=1}^{k}\frac{4^{2kl}×5^{k^{2}}-4^{k^{2}}×5^{2kl}}{4^{l^{2}}×5^{l^{2}}×4^{k^{2}}×5^{k^{2}}}$$
0
votes
2answers
42 views

How this is true $(1-\sum_{i \geq 3}2^{-i}) = (3/4)$?

I'm reading a paper A comparison of two lower bound methods for communication complexity, P.43 $$(1-2^{-1}) \times (1-2^{-2}) \times (1-\sum_{i \geq 3}2^{-i}) = (1/2)(3/4)(1-\sum_{i \geq 3}2^{-i}) = ...
2
votes
2answers
54 views

Why sigma notation?

Repeated union is written as: $$\bigcup_{i=0}^na_i$$ Repeated logical conjunction is: $$\bigwedge_{i=0}^na_i$$ Etc. So why isn't repeated addition: $$\operatorname{\huge+}\limits_{i=0}^n{}^{\Large ...
1
vote
1answer
51 views

Sigma Notations

I have troubles understanding the sigma notation. If for example we have $c_i$ as $$c_i=\frac {x_i-x}{\sum(x_i-x)^2}$$ $$\sum c_i=\sum\frac{x_i-x}{\sum(x_i-x)^2}$$ Do we distribute the sigma to both ...
0
votes
1answer
57 views

How to calculate this sum?

Let $x_1,\cdots,x_k$ be numbers between 0 and 1. Then is it possible to get explicit expression for the following sum:$$\sum_{n_1,\cdots,n_k\geq 1} x_1^{n_1}\times C_{n_1+n_2}^{n_2}\times ...
11
votes
2answers
483 views

Limit of an Integral, Then taking Sum

I am given that $I_n=\int^1_0x^ne^x\,dx$ Now, how can I find the value of the following limit: $$\lim_{n\to\infty}\left(\sum_{k=1}^{n}\frac{I_{k+1}}{k}\right)$$ I suppose solving for $I_n$ is that ...
1
vote
2answers
44 views

Binomial coefficient as a summation series proof?

Alright, so I was wondering if the following is a well known identity or if its existence provides any real benefits other than serving as a time-saver when dealing with higher values for ...
5
votes
4answers
125 views

Surprising Summation (2): $\frac14 \sum_{i=1}^{2n}i(2n-i+1)=\sum_{i=1}^n i^2$

$$\underbrace{\frac14 \sum_{i=1}^{2n}i(2n-i+1)}_{A} =\underbrace{\sum_{i=1}^n i^2}_{B} =\underbrace{\frac 16 n(n+2)(2n+1)}_{C}$$ Transform $A$ directly into $B$ without expanding to $C$ but using ...
1
vote
2answers
47 views

inverting the summation

Let $\{a_n\}$ be a sequence of non-negative real numbers. Then, how can one prove rigoroulsy that $$ \sum_{n=1}^\infty \frac{1}{n^2} \sum_{j=1}^n a_j = \sum_{j=1}^\infty a_j \sum_{n=j}^\infty ...