2
votes
2answers
32 views

How to get sum of points along curve?

startingQuantity / numberAvailable = price 5 apples / 4 available = $1.25 So if there are 4 available and the price is 1.25 and someone buys three apples then ...
0
votes
0answers
55 views

Is this proof correct?

I was working on a summation problem, and I thought of a way to solve it. This is a proof of a generalisation of that method, is it correct? Let $[a,b]_n$ denote the set of a partition of the ...
5
votes
3answers
655 views

Preventing “proof by homework”?

I am doing problem 3d in the Prologue of Spivak: Prove $(a+b)^n = a^n + {n\choose1}a^{n-1}b + {n\choose2}a^{n-2}b^2 + ... + {n\choose n-1}ab^{n-1} + b^n$ I feel like my proof could pass off as ...
2
votes
1answer
74 views

Prove that $\displaystyle\sum_{j=m}^n\sum_{k=0}^{2m}{4j\choose 2k}{2j-k\choose 2m-k}={2n+2m+1\choose 4m+1}2^{4m-1}$

Let $n,m$ are positive integers satisfy the condition $n\ge m>0$ Prove that $\displaystyle\sum_{j=m}^n\sum_{k=0}^{2m}{4j\choose 2k}{2j-k\choose 2m-k}={2n+2m+1\choose 4m+1}2^{4m-1}$
0
votes
0answers
19 views

Summation of a function with the variable both in the function amd in the upper limit

E is defined as : E = c1 ( a$\rho$ + b$\rho ^{2}$ ) + c2 $\rho$ ( c + d $\sum_{j=0}^{n} (\log{ \frac{R\rho}{j} } ) $ ) + c3 $\rho ^{2}$ a, b, c, d, c1, c2, c3, R are known constants. $\rho$ is the ...
0
votes
2answers
21 views

Telescopic summation

I'm working on this question: Rewrite the following summation using sigma notation and then compute it using the technique of telescoping summation. ...
1
vote
2answers
20 views

Compute sum to n

I'm confused as to how to finish answering this question. Compute $$\sum_{i=28}^n (3i^2 - 4i + \frac{5}{7^i}) $$ I end up with $$ 3[\frac{n(n+1)(2n+1)}{6}] - 4[\frac{n(n+1)}{2}] + ...
1
vote
2answers
43 views

Bounding $\sum_{n=n_1}^\infty x^n (n+1)^2$

I need to upperbound the sum $$\sum_{n=n_1}^\infty x^n (n+1)^2$$ where $0<x<1$ is a parameter. I know it can be done starting from $$\sum_{n=n_1}^\infty x^n (n+1)^2\le \sum_{n=0}^\infty x^n ...
1
vote
3answers
63 views

Calculate $\sum_{k=0}^n k \binom{n}{k} p^k (1-p)^{n-k}$

For $p \in [0,1]$ calculate $$S =\sum_{k=0}^n k \binom{n}{k} p^k (1-p)^{n-k}.$$ Since $$ (1-p)^{n-k} = \sum_{j=0}^{n-k} \binom{n-k}{j} (-p)^j, $$ then $$ S =\sum_{k=0}^n \sum_{j=0}^{n-k} k ...
5
votes
2answers
57 views

Question about $ \int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) \forall s\in \Bbb N$

what I found from messing around was $$ \int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) $$ $$ s\in \mathbb{N} $$ when the partial sum is changed to an equivalent polynomial using Faulhaber's formula. ...
2
votes
1answer
67 views

Can someone explain what does this sum mean?

I found a solution to my problem in this thread: How can I (algorithmically) count the number of ways n m-sided dice can add up to a given number? But unfotunately I don't understand the last step. ...
4
votes
1answer
106 views

Using the Parseval Identity to compute $ \sum_{n=1}^{+ \infty} \frac{1}{(4n^2-1)^2}$

Parseval's Identity: For continuous $f: [- \pi , \pi] \to \mathbb{R}$ $$ \sum_{n=- \infty}^{+ \infty} |c_n|^2 = \frac{1}{2 \pi} \int_{ - \pi}^{ \pi} |f(x)|^2dx, \text{ where } c_n = ...
0
votes
1answer
41 views

Prove there is a subsequence $(a_{nk})_{n=1}^\infty$ such that $\Sigma^{\infty}_{k=1} a_{nk}$ converges.

Hey everyone this was give as a practice problem and i'm having trouble, any help is appreciated Let $(a_n)_{n=1}^\infty$ be a sequence such that $\displaystyle \lim_{n \rightarrow \infty} {a_n} = ...
4
votes
2answers
62 views

Is $f(x)=\sum_{k\in\mathbb N}\frac1k\sin\frac x{2^k}$ bounded?

$$f(x)=\sum_{k\in\mathbb N}\frac1k\sin\frac x{2^k}$$Is this function bounded? So obviously this converges because $|\frac1k\sin\frac x{2^k}|<|\frac x{2^k}|$ and $\sum\frac x{2^k}$ converges by ...
1
vote
4answers
80 views

Evaluating $\lim_{n\to\infty} \left({1\over1\cdot2\cdot3}+{1\over2\cdot3\cdot4}+\cdots+{1\over(n-1)\cdot n\cdot(n+1)}\right)$

The original question was to find $L=\displaystyle\lim_{n\to\infty}\sum_{k=1}^na_k$ where $a_n=\displaystyle{n\over(1+2+\cdots+(n-1))(1+2+\cdots+n)}$, which I managed to get down to evaluating the ...
0
votes
1answer
41 views

Prove that if $\sum_{n=1}^\infty a_{n}$ is absolutely convergent, then $|\sum_{n=1}^\infty a_{n}| \leq | \sum_{n=1}^\infty |a_{n}|$

Hey everyone this was given as a practice problem for my first year calculus class and it really giving me a headache, any help is appreciated! Prove that if $\sum_{n=1}^\infty a_{n}$ is absoultley ...
0
votes
1answer
55 views

How to solve this summation (Lerch Transcendent)?

How is it possible to deduce the closed form of the following? $$\sum_{i = 0}^{n - 1} \frac{2^i}{n - i} = ?$$
3
votes
2answers
103 views

Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$

I have been trying to prove this by induction on $n\in \mathbb{N}$, but this approach seemed to get me nowhere. I have a suspicion it might be necessary to express $\log{n}$ as $\int_1^n 1/x\text{ ...
0
votes
0answers
37 views

Way to split up product of summation

If I have $\sum_{n=1}^{\infty}f(x)g(x)$, is there any way to split this up? Thanks.
0
votes
1answer
36 views

Evaluate the following sum

I need explanation how they get to the following equation from left to right considering the partions are $\frac{n}{n}$ $$\sum_{i=1}^n ...
3
votes
1answer
49 views

Cauchy-Schwarz inequality on double-summation term

I have the following, where $v$ is a vector $$ v\cdot (v\cdot \nabla)v $$ which in index notation becomes $v_jv_id_iv_j$. I want to apply the Cauchy-Schwarz inequality on this, which is given by $$ ...
0
votes
3answers
59 views

Using Cauchy-Schwarz inequality to prove that the mean of n real numbers is less than or equal to the root-mean-square of those numbers

Expressed mathematically, the question is to prove the that $\frac{1}{n}$ $\sum_{i=1}^{i=n}{a_i}\leqslant$ $\sqrt{\frac{1}{n}\sum_{i=1}^n{x_i}^2}.$ First of all, what form of Cauchy-Schwarz should I ...
-11
votes
1answer
117 views

1+2+3+4+… = -1/12 [duplicate]

I was browsing through Youtube and I saw this really cool thing.. but it seems really counter-intuitive could someone please explain to me why: 1+2+3+4+..... = -1/12? Here's the link to the guy's ...
1
vote
1answer
65 views

Simplification of a nested sum

I have a nested sum like so: $$\underbrace{\sum_{k_1=k_0}^{k^*} \ ... \sum_{k_n=k_{n-1}}^{k^*}}_{\text{n times}} 1\quad\ \text{with}\ \ n, k_0, k^* \in \mathbb{N},\ k^*\geq k_0$$ Is there a general, ...
0
votes
2answers
87 views

Prove that $f(z) = \sum\limits_{k = 1}^\infty \frac{z^{2^k}}{2^k}$ is continuous in the closed unit disc and holomorphic inside it.

I have started off by assuming that there is a disc of radius $r$ for which $|z|<r$ for $r \in (0,1)$ and $z \in D_r$. This implies that $|z|^{2^k} < r^{2^k}$ And after that, I don't know ...
1
vote
2answers
39 views

Evaluate the equality of two series.

How can we show that these below expressions are not always equal $$\sum_{i=1}^{2}a_{i}b_{i} = \sum_{i=1}^{2}a_{i} \cdot \sum_{i=1}^{2}b_{i} $$ I tried by inputting some numbers but for some reason ...
0
votes
3answers
81 views

Why $\sum_{n=0}^{\infty}(n+1)5^nx^n=\frac{1}{(1-5x)^2}$

Why $$\sum_{n=0}^{\infty}(n+1)5^nx^n=\frac{1}{(1-5x)^2}?$$ I know that $\sum_{n=0}^{\infty}x^n=\dfrac{1}{1-x}$, so by the same token, $\sum_{n=0}^{\infty}5^nx^n=\dfrac{1}{1-5x}$. Thus $$ ...
1
vote
4answers
83 views

Prove that $\sum_{k=0}^{\infty} (k-1)/2^k = 0$

How to prove that this series converges, and that the limit is 0 ?
-1
votes
1answer
73 views

Weird equation - why is that possible?

I have found the following equality: $$\sum_{k=0}^{n} \frac{1}{n+1} = \frac{n+1}{n+1} = 1$$ Why is that possible? I mean the left side is bigger than 1 for any $n>0$. Thank you very much for ...
0
votes
2answers
52 views

Help me come up with a function

I have some numbers and corresponding numbers: 0 = 0 1 = 0 2 = 1 3 = 0 4 = 2 5 = 1 6 = 3 7 = 0 8 = 4 9 = 2 10 = 5 11 = 1 12 = 6 13 = 3 14 = 7 15 = 0 16 = 8 17 = 4 ...
0
votes
2answers
33 views

Shifting Method

I'm taking a Course in Computer Science where we're having a refresher on Calculus 2 material. There is a problem that I don't understand or know how to do. Compute the following using the ...
1
vote
2answers
110 views

$\sum_{r=0}^{\infty} \frac{1}{4r+1} - \frac{1}{4r+3}$

I have no idea how to convert this to an integral(which has to be done as the answer is $\frac{\pi}{4}$) I assume it may be equivalent to arctan(1).
2
votes
1answer
62 views

Help show that a second derivative is always negative

How do I show that the second derivative is always negative? I've computed the second derivative to be: $\displaystyle\frac{n}{2\sigma^4}-\frac{1}{\sigma^6}\sum\limits_{i=1}^n(x_i-\mu)^2$ Then I ...
0
votes
1answer
20 views

Backsolving power curve with uplift

We have a curve $cx ^ b$. Let's say I sum $x$ 1 through to 52 using that curve and it gives me a number, say 10. I want to increase number 10 by 10% and we know that's 11. I can increase 10% very ...
1
vote
0answers
39 views

Multiple equations with relations.

I'm very bad at math so please correct me or let me know if I'm posting in the wrong section. I have a set of equations (in reality any number > 2) but if its solvable for 3 it would be good enough. ...
3
votes
1answer
110 views

Evaluating an infinite summation

The question is to evaluate this: $$ \lim_{n\to\infty} \sum_{r=0}^{n}\left(\dfrac{1} {4r+1} - \dfrac{1}{4r+3}\right) $$ The hint given is that, the above is equal to: $$ \int_0^1((1 + x^4 + x^8 + ...
3
votes
1answer
475 views

Is this infinite sum always less than zero?(+500pts bounty for the correct answer)

I wonder if the following infinite sum is always negative for all (finite) $A,d>0$ and $B<0$. Any counterexample also suffice. Here is the sum: $$\frac{\partial}{\partial d}\sum_{n=1}^\infty n ...
0
votes
0answers
14 views

Order of a sum, polynomial and exponential terms

I would like to understand if the following sum, $$\sum_{x=0}^L x^{d-1} p ^ {L-x}$$ is of order $L^{d-1}$ or of order $L^{d}$. In the previous expression $0<p<1$ and $d$ is an integer. In ...
10
votes
3answers
446 views

an inequality: $1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53$

$n$ is a positive integer, then $$1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53.$$ please don't refer to the famous $1+\frac1{2^2}+\frac1{3^2}+\dotsb=\frac{\pi^2}6$. I want to find a ...
5
votes
3answers
143 views

How find this sum of $\lim_{n\to\infty}\sum_{k=1}^{n}\left(\frac{1}{2^k}\sum_{i=0}^{2^{k-1}-1}\ln{\left(\frac{2^k+2+2i}{2^k+1+2i}\right)}\right)$

Find the sum of the limit $$\lim_{n\to\infty}\sum_{k=1}^{n}\left(\frac{1}{2^k}\sum_{i=0}^{2^{k-1}-1}\ln{\left(\frac{2^k+2+2i}{2^k+1+2i}\right)}\right)$$ My try: since ...
1
vote
6answers
213 views

Evaluating $\lim_{n\to \infty } \, \left(\sum _{k=1}^{\infty } \frac{1}{n}\right)$

$$\lim_{n\to \infty } \, \left(\sum _{k=1}^{\infty } \frac{1}{n}\right)$$ Intuitively, it seems that you are adding infinite of $\frac{1}{n}$ and then taking the limit as n goes to infinity, which ...
2
votes
2answers
96 views

Finding the complex fourier series of the function $x^2sin(x)$ in the interval $[{-\pi}, \pi]$?

This forms part of a project I am doing and I wish to see how well complex fourier series approximates a smooth curve such as this one. After tedious integration by parts, I have attained an answer ...
5
votes
1answer
101 views

Simplify $\sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x)$

Simplify the following expression $$S_N = \sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x), $$ where $a$ is a real number and $f(x)$ is an analytic real function. What is $\lim_n ...
0
votes
3answers
148 views

Show that the infinite sum converges. [closed]

$$\sum_{n=1}^{\infty}\frac{(-1)^{(n-1)} } {n^x}$$ Show that the series converges.
5
votes
3answers
272 views

Evaluating $\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$

How can we obtain following formula? $$\sum_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}=\frac{1}{3}e^{\frac{-x}{2}}x\left(e^{\frac{3x}{2}}-2\sin\left(\frac{\pi+3\sqrt{3}x}{6}\right)\right).$$ I think if we ...
0
votes
0answers
21 views

Vector representation of a sum

I am trying to solve a minimization problem and I encountered the following sum: $\sum_i^n x_i^4$ I know that $\sum_i^n x_i^2 = x'x$ but I can't figure out what the vector representation is of the ...
1
vote
1answer
27 views

Comparison of $S_{n}$ and $T_{n}$, where $S_{n} = \sum_{k=1}^{n}\frac{n}{n^2+nk+k^2}$ and $T_{n} = \sum_{k=1}^{n-1}\frac{n}{n^2+kn+k^2}$

Let $\displaystyle S_{n} = \sum_{k=1}^{n}\frac{n}{n^2+nk+k^2}$ and $\displaystyle T_{n} = \sum_{k=1}^{n-1}\frac{n}{n^2+kn+k^2}$ for $n=1,2,3,\dots$ Then which of the following options are Right. ...
3
votes
1answer
44 views

power sum of $\sum_{n=1}^\infty 2^n x^{n^2}$

How do I calculate the radius of the power sum of: $$\sum_{n=1}^\infty 2^n x^{n^2}$$ having a little trouble with the second power(of $x$) thank you
2
votes
3answers
74 views

Prove that $2\sum_{k=1}^n \cos(kθ) = \frac{\sin[\left(n+1/2\right)θ]}{\sin(θ/2)}-1$ [closed]

Prove that $$2\sum_{k=1}^n \cos(kθ) = \frac{\sin[\left(n+1/2\right)θ]}{\sin(θ/2)}-1$$ By using $$e^{iθ}+e^{2iθ}+\cdots+e^{niθ}=\frac{e^{iθ}(1-e^{inθ})}{1-e^{iθ}}$$
1
vote
1answer
27 views

Summation for computing distance with time?

I have a homework problem for Calc II that goes something like this: A tortoise and a hare are in a 1500m race. The hare goes 1m in the first second, (999/1000)m in the second second, a (999/1000)^2 ...