# Tagged Questions

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### 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!
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### 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 ...
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### 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.
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### 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 ...
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### 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?
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### 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 ...
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### 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
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### How large should $a$ be so that $\int_a^{\infty} \frac{dx}{1+x^2} < \frac{1}{1000}$

I want to solve this without using calculator.
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### 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 ...
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### 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 ...
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### 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 ! :)
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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.
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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$$\dfrac{1}{n^5-5n^3+4n} = ... 2answers 35 views ### If a sequence \{a_n\} satisfies the Inequality a_{n+1} < ka_{n}, then show that  \lim\limits_{n \to \infty} a_n =0 where 0< k , a_n< 1 I know one solution. Consider \sum a_n Then use ratio test to show that the series converges, hence the sequence. Any other Ideass ! 3answers 71 views ### Find all values of c for which the following series converges \sum_{n=1}^{\infty} \left(\dfrac{c}{n}-\dfrac{1}{n+1}\right) I know that the series in question converges when c=1, but I have no concrete way to find all such values of c for which this is true. 1answer 56 views ### Give example of a series \sum a_n such that the series is conditionally convergent. and \sum na_n is convergent I tried all the conditionally convergent series I know, I found \sum na_n to be diverging for all of them. But I am sure the question is correct 3answers 322 views ### Proving that  \displaystyle \gamma = \int_{0}^{1} \!\!\int_{0}^{1} \!\frac{x - 1}{(1 - x y) \log(x y)} \, \mathrm{d}{x} \, \mathrm{d}{y} . In 2005, J. Sondow found a surprising formula for the Euler-Mascheroni constant  \gamma . The formula is$$ \gamma = \int_{0}^{1} \int_{0}^{1} \frac{x - 1}{(1 - x y) \log(x y)} ~ \mathrm{d}{x} ~ ...
Prove that $$\sum_{i=1}^{\infty} \frac{1}{i(2i+3)} = \frac89 -\frac23\ln2$$ I tried using integration but failed miserably. Hints please.