1
vote
2answers
35 views

Calculus add formula to derive new formula

I was asked to re-write a formula forward and backward and derive a new formula from it. Here's the problem: Here us formula 5.1.4: I'm not too sure where to start. Thanks!
1
vote
1answer
60 views

Summation of: $\sum_{1}^{\infty}\left(\frac{2}{3}\right)^x$ [duplicate]

This is a subsection in my statistics homework. It goes back to calculus II and summations, and it's been a long time since I've studied it so I'm rusty. I'm looking to solve the summation of ...
1
vote
4answers
41 views

Why is $\sum\limits_{b=1}^{t-1} {t \choose b} 2^{t-b} = (3^t - 2^t - 1)$

Why is $$\sum\limits_{b=1}^{t-1} {t \choose b} 2^{t-b} = (3^t - 2^t - 1)$$ Thanks.
1
vote
1answer
63 views

Derive the formula for the sum of the first $n$ squares using derivatives and integrals

I wanted to prove the formula for sum of squares without using induction and thought using derivatives would be the easiest approach ...
17
votes
4answers
395 views

A sum containing harmonic numbers $\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n}$

I'm trying to find a closed form for the following sum $$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$ where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number. Could you help me with it?
2
votes
2answers
57 views

Show that $H_i=H_{n-i}$ and $\sum H_i=1$

We define $$H_i=\frac{1}{n}\frac{(-1)^{n-1}}{i!(n-1)!}\int_{0}^{n}\prod_{j=0,j\neq i}^{n}(x-j)dx$$ This is called the Newton-Cotes coefficient. Here is the exercise: First, convince yourself that ...
2
votes
2answers
77 views

How to prove $n! > n^a$ for all $a\in \mathbb{R}$ (for sufficiently large $n$)?

I've encountered a proof which claims $n! > n^2$ for sufficiently large $n$. I tried using induction to prove it for an arbitrary $a$: $n! > n^a$. Lets assume the claim is true for $n$: $n! ...
0
votes
1answer
37 views

Prove the serie is bounded by $a_{m+1}$ for all $m\in \Bbb{N}$

Let $a_n$ a monotone sequence approaches $0$. Show that for all $m\in\Bbb{N}$: $$ 0 < (-1)^m\sum\limits_{n=m+1}^{\infty} (-1)^{n+1} {a_n} < a_{m+1} $$ I wanna focus on the RHS inequality: ...
2
votes
1answer
44 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
4
votes
6answers
733 views

What is the limit of the following sum

$$\lim_{n\to\infty}\sum_{k=1}^n \ln\Big(1+\frac{k}{n^2}\Big)$$ According to me, the answer is $0$. I'm curious as to what answers might others come up with, as well as the method of reasoning.
8
votes
0answers
71 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
4
votes
2answers
183 views

Integral of the Von karman equation

What is the result of this integral, and how can I proceed: $$ \int_{-\infty}^{\infty}{c_{1} \over\left(1 + c_{2}\,x^{2}\right)^{5/6}}\, \cos\left(x\tau\right)\,{\rm d}x\,,\qquad c_{1}, ...
0
votes
3answers
38 views

Solving the Riemann Sum $\sum_{i=1}^{n}(1+\frac{6i}{n})^3(\frac{2}{n})$?

So I have the Riemann sum. $\sum_{i=1}^{n}(1+\frac{6i}{n})^3(\frac{2}{n})$. From my understanding that turns into $(\frac{2}{n})\sum_{i=1}^{n}(1+\frac{6i}{n})^3$ and what is really perplexing me is ...
2
votes
2answers
45 views

Help finding a summation using CAS

While approximating an integral by midpoint rule, I ended up with $$\iint_R f(x, y)\hspace{1mm}dA\approx \dfrac{1}{n^2}\sum_{i=0}^{n-1}\sum_{j=0}^{n-1} f\left[\dfrac{1}{2n}+\dfrac{i}{n},\hspace{3mm} ...
1
vote
1answer
55 views

The Summation of x(log(log(x))

So I would like to know if it is possible to express this summation in terms of $n$: $$\sum_{x=2}^n x\log(\log(x))$$ For example the summation below is equivalent to $\frac12 n (n+1)$ in terms of ...
0
votes
2answers
32 views

How do I expand this summation?

So I just started doing these today and this has me stuck (It's a beginner question and i'm upset I'm stumped). So I have $\sum_{k=1}^{4}9k\sin(\frac{k\pi}{2})$ which I turn into ...
0
votes
2answers
62 views

Find a sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$

Find a sum of $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$$ Could you give some some hint or some way to start this? I have tried representing ch(n) through its definition with e, but I ...
4
votes
4answers
762 views

Why don't we indicate the variable to summed as we do for integrals?

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so: $$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes ...
0
votes
3answers
51 views

$\sum_{(p,q) \in {\mathbb{N}^*}^2 and p \land q =1} \frac{1}{p^2 q^2} = \frac{5}{2}$ proof? [closed]

Can you give me a very precise demonstration of this result please because it's very difficult for me to understand the demonstration on the pic :( $$ \sum_{(p,q) \in {\mathbb{N}^*}^2 \text{, } p ...
0
votes
1answer
25 views

Simplify $S=\sum_{i=0}^{k}a_i (2n)^{2i+1}$

Can someone simplify this expression (or compute its supremum)? Thanks so much. $$S=\sum_{i=0}^{k}a_i (2n)^{2i+1}$$ where $a_i>0$ and $k>1$, and $\sum_{i=0}^{k}a_i=1$.
0
votes
2answers
67 views

Compound interest with a compounding interest rate

I have an investment which pays 3% interest (r) annually but it also increases the interest rate every year by 5% (g). I re-invest all interest payments at the start of each year. How many years (t) ...
7
votes
3answers
591 views

How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$?

How to find the value of $S(\infty)$, where $S(n)$ is the following $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$ Wolfram alpha is unable to calculate it. This is a question from a ...
7
votes
4answers
161 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
1
vote
2answers
100 views

How to find $\sum_{r=1}^{n} r^2\cos {(r\theta)}$

How do you find the sum $$S(\theta)=\displaystyle\sum_{r=1}^{n} r^2\cos {(r\theta)}$$? I observed that if $f(\theta)$ is $\sum\cos {(r\theta)}$, then $S(\theta)=-f''(\theta)$. Help will be ...
0
votes
2answers
25 views

How to get a partial sum formula

Let S denote sum from 1 to n of (k-1)/k! . I tried obtaining a partial sum formula, but I couldn't get too far. WolframAlpha comes with quite a simple form, but I fail to see how they got there . Can ...
3
votes
1answer
50 views

investigate $\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$

I need to investigate the series (Hence, when the series converges and when the series converges absolutely depending on $\alpha$). $$\sum\limits_{n\ge2}{\frac{(-1)^n}{n^\alpha \ln n}}$$ For ...
1
vote
1answer
31 views

evaluating a sum using Cauchy condensation test

Let $$\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$$ I want to check if the sum is converges absolutely. Hence, we need to check the convergence of $$\sum\limits_{n\ge1}{\frac{1}{n^\alpha \ln ...
6
votes
2answers
196 views

What is the closed form for $S=\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ({n})}{n!}$?

How do we find the following sum (closed form)? $$S=\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ({n})}{n!}$$
0
votes
2answers
65 views

How to find the value of this summation?

How to solve this summation? $$S=\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n4^{4n+1}}$$ I tried it to convert it into a definite integral but wasn't successful. Help will be appreciated.
10
votes
3answers
196 views

How to find sums like $\sum_{k=0}^{39} \binom{200}{5k}$

How do I find sums like these?-- $$S=\displaystyle\sum_{k=0}^{39} \dbinom{200}{5k}$$ that is, when there is a summation of binomial coefficients, but with jumps of some terms..?
3
votes
1answer
31 views

Inequality containing finite sum.

For what value of k the following inequality holds? $\sum_{i=1}^{n}a_{i}^3<k|\sqrt{\sum_{i=1}^{n}a_{i}}|$ I don't have any idea to solve this.
4
votes
3answers
494 views

Does a closed form exist for this summation?

How do I calculate $$\sum_{k=1}^\infty \frac{k\sin(kx)}{1+k^2}$$ for $0<x<2\pi$? Wolfram alpha can't calculate it, but the sum surely converges.
2
votes
3answers
45 views

Series question with logarithms

I want to know how to check the divergence of following sum: $\sum_{k=0}^\infty \frac{1}{\sqrt[n]{\log n}}$ I tried to use this result: $ \lim_{n \rightarrow \infty} \frac{1}{\sqrt[n]{\log n}}=1 ...
4
votes
6answers
106 views

If $\lim\limits_{x \to \infty} f(x) = 1$, can we have function $f(x)$, such that $\int_0^{\infty}f(x)dx$ converges

I know the Initiative answer, can anyone give a neat answer based on solid reasoning EDIT : $f(x)$ is continuous
0
votes
4answers
138 views
0
votes
0answers
38 views

Taking the derivative of a summation

I need to know how to solve this, I would prefer a step by step process, and not just a solution please, working with perceptrons in a neural net. n is the number of nodes in a layer, every node is ...
3
votes
1answer
64 views

Can a general formula be developed for these integrals?

Can the following integral $I$ be written wrt $n$? $$I=\displaystyle\int_{0}^{\infty} \dfrac{dx}{\sum_{k=0}^{n} x^k}$$ I found the values for $n=1,2,3$ but can we generalize it for an arbitrary ...
2
votes
6answers
221 views

Fastest way to integrate $\int_0^1 x^{2}\sqrt{x+x^2} \hspace{2mm}dx $

This integral looks simple, but it appears that its not so. All Ideas are welcome, no Idea is bad, it may not work in this problem, but may be useful in some other case some other day ! :)
2
votes
1answer
82 views

What is the inverse function of $\int{ \frac{1}{{\sqrt{x+1}}{x^n}} dx}$?

I am trying to solve $$ \frac{dy}{dt} = \alpha ((y+1)^2 - \gamma)^n \hspace{2cm} y(0)=0 $$ Here $y$ is a real-valued, monotonically increasing, positive definite function of $t$ in the interval ...
1
vote
3answers
72 views

Can you show that the LHS equals the RHS in this equation, by showing how I can get the expression on the RHS?

$$ \frac{1^2+2^2+...+(n-1)^2}{n^3} = \frac{(n-1)n(2n-1)}{6n^3} $$ Can someone show me step by step how I can transform the LHS to the RHS? If possible, using high school-level math. I have now ...
1
vote
1answer
46 views

Is an integral without a differential component on a finite number of points just a sum?

Is an integral $$\int_{\lbrace 1, 2, 3 \rbrace} f(x)$$ simply the sum $$\sum_{x=1,2,3} f(x)?$$ I ask this question because of the generalization to multiple dimensions of integration by parts ...
-1
votes
1answer
72 views

Rewrite and approximate the sum as an Integral $\sum_{i=1}^{1000} \sqrt{i}$ [closed]

This is not an Infinite sum !, how do we change this to an Integral. $ $ We normally write an integral as an infinite sum.
5
votes
1answer
63 views

How many decimal representations are possible for the number 1

I know that there at least two $0.\overline{9}$ and 1 Is there a neat and more concrete way to understand this problem.
1
vote
1answer
137 views

Find the sum of the series $\sum \limits_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$

Feel free to skip obvious steps, or use a calculator when required. I just want to understand the theme of the solution. Any help is appreciated EDIT : We can write$$ \dfrac{1}{n^5-5n^3+4n} = ...
0
votes
2answers
35 views

If a sequence $\{a_n\}$ satisfies the Inequality $a_{n+1} < ka_{n}$, then show that $ \lim\limits_{n \to \infty} a_n =0$ where $0< k , a_n< 1$

I know one solution. Consider $\sum a_n$ Then use ratio test to show that the series converges, hence the sequence. Any other Ideass !
1
vote
3answers
71 views

Find all values of $c$ for which the following series converges $\sum_{n=1}^{\infty} \left(\dfrac{c}{n}-\dfrac{1}{n+1}\right)$

I know that the series in question converges when $c=1$, but I have no concrete way to find all such values of $c$ for which this is true.
3
votes
1answer
56 views

Give example of a series $\sum a_n$ such that the series is conditionally convergent. and $\sum na_n$ is convergent

I tried all the conditionally convergent series I know, I found $\sum na_n$ to be diverging for all of them. But I am sure the question is correct
12
votes
3answers
322 views

Proving that $ \displaystyle \gamma = \int_{0}^{1} \!\!\int_{0}^{1} \!\frac{x - 1}{(1 - x y) \log(x y)} \, \mathrm{d}{x} \, \mathrm{d}{y} $.

In 2005, J. Sondow found a surprising formula for the Euler-Mascheroni constant $ \gamma $. The formula is $$ \gamma = \int_{0}^{1} \int_{0}^{1} \frac{x - 1}{(1 - x y) \log(x y)} ~ \mathrm{d}{x} ~ ...
4
votes
3answers
169 views

Infinite Sum of algebraic expression

Prove that $$\sum_{i=1}^{\infty} \frac{1}{i(2i+3)} = \frac89 -\frac23\ln2$$ I tried using integration but failed miserably. Hints please.
2
votes
0answers
35 views

statements about summation

Could you help me prove this statements about summation? I know that the second prove is easy of be written, but can I put that summation before cos(theta) and sin(theta)? yes. But why? Do you ...