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

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8
votes
2answers
287 views

Can we add an uncountable number of elements, and can this sum be finite?

Can we add an uncountable number of elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. Any help ...
5
votes
2answers
118 views

What is the sum of this series? $\sum_{n=1}^\infty\frac{\zeta(1-n)(-1)^{n+1}}{2^{n-1}}$

I want to know what is the sum of this series: $$\sum_{n=1}^\infty\frac{\zeta(1-n)(-1)^{n+1}}{2^{n-1}}$$ $B_n$ are the Bernoulli numbers. Mathematica does not help.
0
votes
1answer
49 views

What is the sum of this series? $\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$ [duplicate]

I want to know what is the sum of this series: $$\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$$ Mathematica does not help.
0
votes
2answers
43 views

Can absolute value functions be moved like this?

If I have an expression that looks like $|x-a_1| + |x-a_2| + |x-a_3| + ... + |x-a_n|$ Is it the same as doing $|nx - \sum_{i=1}^{n}a_i|$
2
votes
0answers
61 views

The sum of $\frac{k^{2}}{k-1}$ from 6 to 12

Can you evaluate this sum by using the properties of the sigma notation ? Or I must develop this and evaluate them one by one ?
2
votes
1answer
49 views

How many terms required in $e =\sum^∞_{k=0}{1\over k!}$ to give $e$ with an error of at most ${6\over 10}$ unit in the $20$th decimal place?

How many terms are required in the series $e =\sum^∞_{k=0}{1\over k!}$ to give $e$ with an error of at most ${6\over 10}$ unit in the $20$th decimal place? Here is what I have: $$e\approx ...
0
votes
0answers
43 views

tricky derivative with logarithm of sum

I'm having trouble understanding the solution of a limit. It involves a formula for measuring certainty of a discrete probability distribution. Given a set of values $p_j$ which sum up to 1, find the ...
1
vote
1answer
142 views

Why is $1+2+3+\cdots = 0 $? [duplicate]

I had seen this result a while back in a Numberphile video: $1+2+3+\cdots = -\frac{1}{12}$ I was trying to prove the same result using a different method when I accidently proved that the sum was ...
-5
votes
0answers
31 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
6answers
58 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!
2
votes
2answers
59 views

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

$$\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$$ 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 ...
2
votes
2answers
84 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
38 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 $n,X(0),Y(0)$. $$ X(n) = X(n-1) + Y(n-1) \\ Y(n) = 2X(n-1) + Y(n-1) $$
4
votes
3answers
69 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
27 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
37 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
156 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
19 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
51 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
34 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
127 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 ...
-4
votes
1answer
43 views

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

How can I calculate this, where $c>0$? $$ \sum_{r=k}^{\infty} \frac{e^{\frac{-c k}{2 r}}}{r^{3/2}} $$ One solution might be to upper bound and lower bound this sum by an integral, and then take ...
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
137 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
46 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 ...
5
votes
4answers
126 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
62 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
129 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
37 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
1answer
82 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
40 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: $$S=\sum_{k=1}^{n-1}\sum_{l=1}^{k}\frac{4^{2kl}×5^{k^{2}}-4^{k^{2}}×5^{2kl}}{4^{l^{2}}×5^{l^{2}}×4^{k^{2}}×5^{k^{2}}}$$