All Questions
81
questions
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2
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212
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Evaluate the sum by writing it as a definite integral
I am asked to evaluate the following sum by writing it as a definite integral. I was hoping if my methods we correct for the following sums.
$\displaystyle{\lim_{n \to \infty} \frac{3}{n}\sum_{i=1}^{...
- 2,907
1
vote
1
answer
105
views
Why is $\sum a_nf(n) = \int_0^xf(t)~d(A(t))$?
This equation is a part of Abel's summation formula
$a_n$ is a sequence, $f$ is a real differentiable function such that $f'$ is Riemann integrable.$$A(x) = \sum_{1\le n\le x}a_n$$
I don't see why is $...
- 1,641
0
votes
1
answer
113
views
Integration as a limit of sum.
Let $\displaystyle f(x)=\lim_{n\to\infty} \left(\frac{ax}{n}\left(\sum_{k=1}^n \frac{[k^2-e^{-x}+k-1]}{k(k+1)}\right)\right)+\lambda$
Find $f(x)$ if $[\,.]$ denotes G.I.F.
I know how to solve such ...
- 3,030
2
votes
1
answer
409
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Comparing summation and integration for non monotonic function
$$P=\sum_{r=3n}^{4n-1} \frac{r^2+13n^2-7rn}{n^3}$$.
$$Q=\sum_{r=3n+1}^{4n} \frac{r^2+13n^2-7rn}{n^3}$$.
$$I=\int_{3}^{4} (x^2-7x+13) dx = \frac{5}{6}$$
Compare the values of $P,Q,I$
I know ...
- 2,205
0
votes
2
answers
144
views
Approximate the integral $\int_0^{0.5}{x^2e^{x^2}}dx$ correct to four decimal places using a Maclaurin series.
I got $$\int_0^{0.5}{\sum_0^\infty}\frac{x^{2n+2}}{n!}dx$$ for the taylor series representation, but I'm not sure what to do next.
Do I use 0 and 0.5 as bounds for z for the Lagrange Error Bound? And ...
user738554
0
votes
3
answers
269
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How to evaluate $\int_0^1 -\frac {\ln(1-x)}{x}dx$
This integral came up while I was trying to evaluate $$\sum_{n=1}^\infty \frac {1}{n^2}.$$ The value should be $$\frac {\pi^2}{6},$$ but how do I solve the integral and evaluate it?
- 137
32
votes
1
answer
818
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On the relationship between $\Re\operatorname{Li}_n(1+i)$ and $\operatorname{Li}_n(1/2)$ when $n\ge5$
Motivation
$\newcommand{Li}{\operatorname{Li}}$
It is already known that:
$$\Re\Li_2(1+i)=\frac{\pi^2}{16}$$
$$\Re\Li_3(1+i)=\frac{\pi^2\ln2}{32}+\frac{35}{64}\zeta(3)$$
And by this question, ...
- 8,679
3
votes
0
answers
226
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integrability question, lower\upper integral and the lower\upper Riemann sum
Define a function $g : [0, 1] \mapsto \mathbb R$ by the following formula:
$$g(x)=\begin{cases}-1,&x\in \mathbb{Q}\\x^3-x,&x\not\in \mathbb{Q}\end{cases}$$
(a) What is $\underline{I}^...
- 5,283
8
votes
3
answers
407
views
An integration-via-summation formula
For symbolic transformation of integrals and series I occasionally use this formula:
$$\int_0^1f(x)\,dx=-\sum_{n=1}^\infty\sum_{m=1}^{2^n-1}\frac{(-1)^m}{2^n}f\left(\frac m{2^n}\right)\tag{$\diamond$}$...
- 47.3k
0
votes
2
answers
63
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The limit $n \rightarrow \infty$ of the standard deviation of $x_k= \ln k, k=1,2,3,..,n.$
The standard deviation for a sequence $x_k$ is defined as
$$S_n=\sqrt{\sum_{k=1}^{n} \frac{{x_k}^2}{n}-\left(\sum_{k=1}^{n} \frac{x_k}{n}\right)^2}$$
By numerics the asymptotic value of $S_n$ for $x_k=...
- 43.6k
4
votes
3
answers
66
views
How to solve $\lim_{n \to \infty} \overset{n}\sum_{k=1} 3(1+\frac{2k}{n})\frac{2}{n}$?
I need to solve $\lim_{n \to \infty} \overset{n}\sum_{k=1} 3(1+\frac{2k}{n})\frac{2}{n}$
I have worked to this by trying to calculate $\int^3_{1}3xdx$, but I am not sure how to get rid of the $k$ and ...
- 1,661
0
votes
0
answers
63
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Find the limit when $n \rightarrow \infty$ of the series
Find the limit when $n \rightarrow \infty$ of the series:
$$\frac n{n^2}+\frac n{n^2+1^2}+ \frac n{n^2+2^2}+\cdots+\frac
1{n^2+(n+1)^2}$$
I am required to do this using limit of a sum definition ...
- 1,147
2
votes
0
answers
107
views
Why are integrals and summations useful in computer science and what do these function mean?
I am reading my way through an introductory MIT computer science book called Structure and Interpretation of Computer Programming, and while I understand the programming and logic behind the book, ...
- 231
5
votes
1
answer
4k
views
Summation of series using definite integral
I learned that definite integral gives the signed area under a curve by dividing the curve into small rectangular strips and "making" its width shrink to zero.
Using this knowledge summation of ...
- 247
1
vote
1
answer
133
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Edit: Solved, Calculus homework: rewriting limit as definite integral
On my Calculus homework, there is a question that I am having trouble with.
The question is:
Rewrite
$\lim_{n \to \infty}\sum_{i=1}^{n}\frac{1}{n}\left[\left(\frac{i}{n}\right)^{3}+\left(\frac{i}{n}\...
