All Questions
81
questions
1
vote
1
answer
93
views
Evaluating the infinite sum $\lim_{n \to \infty}\sum_{r=0}^n\frac{1}{2r^2 +3r+1}$
How do I evaluate :
$\lim_{n \to \infty}\displaystyle \sum_{r=0}^n\left(\dfrac{1}{2r^2 +3r+1}\right)$
I tried sandwhich theorem and even wrote down some terms to see any particular pattern but ...
- 6,111
8
votes
1
answer
251
views
On the integral $\int_1^\infty\big(\{x\}^n-\frac1{n+1}\big)\frac{dx}x$
According to Dirichlet's test (integral version),
$$
I_n=\int_1^\infty\big(\{x\}^n-\frac1{n+1}\big)\frac{dx}x
$$
converges, where $n$ is a positive integer and $\{x\}$ denotes the fractional part of $...
- 8,679
0
votes
1
answer
65
views
Finding upper sum using $n$ sub intervals
enter image description hereI am asked to calculate the upper sum of $f(x)= 5-2x$, from $x=1$, to $x=2$ using $n$ subintervals.
Below is my working out, however, the answer is $2 + \frac1n$.
I will ...
- 327
3
votes
0
answers
253
views
Finding upper and lower sum of a Heaviside function.
I have been asked to find the lower and upper sum of a Heaviside function. It is a combination of a Heaviside function, like: H(x-1) + (3x-12)H(x+1).
I have been able to draw the graph for this, ...
- 89
1
vote
7
answers
986
views
Why is it that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ is not less than $\int_1^\infty \frac{dx}{x^2} = 1$?
So according to Euler's proof of the Basel problem,
$$\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6},$$
But only for $n \in \mathbb{Z}$.
But if $n$ was a positive real and $n \geqslant 1$, then ...
- 860
4
votes
2
answers
182
views
Prove that $\sum^m_{k=0}\binom{m}{k}(-1)^k(\frac{1}{n+k+1}) = \frac{m!n!}{(m+n+1)!}$
I am looking for a more direct proof of the following identity: $$\sum^m_{k=0}\frac{\binom{m}{k}(-1)^k}{n+k+1} = \frac{m!n!}{(m+n+1)!}$$
My original proof comes from evaluating $\int^1_0{x^n(1-x)^m}{...
- 786
1
vote
2
answers
217
views
Sophomore's Dream : integral not defined in x=0
Sophomore's dream is the identity that states
\begin{equation}
\int_0^1 x^x dx = \sum\limits_{n=1}^\infty (-1)^{n+1}n^{-n}
\end{equation}
The proof is found using the series expansion for $e^{-x\...
- 11
-2
votes
2
answers
72
views
What is the limit of $\sum_{k=1}^{n}\frac{k^3}{n^4}$? [closed]
Find the following limit:
$$\lim_{n \to \infty} \sum_{k=1}^{n}\frac{k^3}{n^4}$$
Should I approximate this with integrals?
- 23
4
votes
1
answer
162
views
Calculate the limit of integral sum
Find limit of $$a_n = \sum \limits_{k = 1}^n \frac{2^{\frac{k}{n}}}{n + \frac{1}{k}}$$
My attempt is:
$$\sum \limits_{k = 1}^n \frac{2^{\frac{k}{n}}}{n + \frac{1}{n}} < \sum \limits_{k = 1}^n \...
- 151
9
votes
3
answers
2k
views
Evaluating $\int_0^1 \frac{\ln^m (1+x)\ln^n x}{x}\; dx$ for $m,n\in\mathbb{N}$
Evaluate $\displaystyle \int\limits_0^1 \dfrac{\ln^m (1+x)\ln^n x}{x}\; dx$ for $m,n\in\mathbb{N}$
I was wondering if the above had some kind of a closed form, here some of the special cases have ...
- 2,826
8
votes
1
answer
317
views
Evaluating the integral $\frac{1}{2^{2n-2}}\int_0^1\frac{x^{4n}\left(1-x\right)^{4n}}{1+x^2} dx$
Prove that :
$$ \frac{1}{2^{2n-2}}\int \limits_{0}^{1} \dfrac{x^{4n}\left(1-x\right)^{4n}}{1+x^2} dx =$$$$\sum \limits_{j=0}^{2n-1}\dfrac{(-1)^j}{2^{2n-j-2}\left(8n-j-1\right)\binom{8n-j-2}{4n+j}} + (-...
user399078
2
votes
1
answer
142
views
An integral transformed to arctan series
I wanted to calculate this one$\displaystyle\int\limits_{0}^{\infty} \dfrac{\sin 2x}{x(\cos x+\cosh x)}\; dx$
I used $\displaystyle 2\sum\limits_{k=1}^{\infty}(-1)^{k-1}\sin kx e^{-kx}=\dfrac{\sin x}{...
- 2,826
1
vote
2
answers
179
views
Evaluation of $\sum_{n=1}^{\infty} (\frac{1}{4n-3}-\frac{1}{4n-1}) $
Prove that
$$\sum_{n=1}^{\infty} (\frac{1}{4n-3}-\frac{1}{4n-1})=\frac{\pi}{4}$$
Could someone give me slight hint as how to convert above expression to $\int_{0}^{1} \tan^{-1}x.dx$
- 1,110
1
vote
2
answers
128
views
$\lim_{n \rightarrow \infty} \sum_{n=1}^{n}\frac{a_n}{n}$ is equal to?
Let $$a_n=\int_{0}^{\pi/2}(1-\sin(t))^{n}\sin(2t)\, dt$$ then $$\lim_{N
\rightarrow \infty} \sum_{n=1}^{N}\frac{a_n}{n}$$ is equal to?
I tried to apply L'Hospital's rule initially but it will not ...
user220382
1
vote
1
answer
475
views
Is there a closed form expression for the following lower and upper Riemann Sums?
Problem: Consider partitioning $f(x)=x^2$ on $[-1,1]$ into $n$equal sub-intervals. We seek to derive an expression for $L(f, P)$ and $U(f, P)$ in terms of $n$, where $n$ is even.
Now the question ...
- 13.2k