All Questions
47
questions
1
vote
0
answers
50
views
Area under a curve using the rectangle method
Take $$x_k$$ as the left endpoint of each subinterval to find the area under the curve y = f(x) above the specified interval.
f(x) = 9 − x^2; [0, 3]
What I've done so far is to consider $$\Delta x = \...
2
votes
2
answers
68
views
How to show that a sum is inclosed by two values / bounds?
Show that this holds for all c>0
$\frac{\pi}{2\sqrt{c}} \le \sum_{n=0}^{\infty} \frac{1}{n^2 + c} \le \frac{\pi}{2\sqrt{c}} + \frac{1}{c}$
I'd really appreciate it if some can tell me if my proof ...
- 1,237
1
vote
0
answers
59
views
How do I prove that this finite sum evaluate to $1+\sqrt{2}$ for all values of n? [duplicate]
$$
\frac{\displaystyle\sum_{i=1}^{n-1}
\sqrt{\sqrt{n} + \sqrt{i}}}
{\displaystyle\sum_{i=1}^{n-1}
\sqrt{\sqrt{n} - \sqrt{i}}}
= 1 + \sqrt{2}
$$
When $n=2$, this is easy to verify. As $n \to \infty$...
- 1,240
1
vote
1
answer
78
views
Is the statement $\sum_{j=1}^\infty x_j<\infty,~(x_j\ge0)$ $\Longrightarrow \lim _{k \to \infty} \sum_{j=k}^\infty x_j=0$ true?
As the title states, I would like to know if the statement
$$
\sum_{j=1}^{\infty} x_{j}<\infty \Longrightarrow \lim _{k \to \infty} \sum_{j=k}^{\infty} x_{j}=0,\qquad x_j\in [0,\infty)
$$
is always ...
- 853
0
votes
1
answer
89
views
Limit of a Riemann Sum. [duplicate]
I am trying to calculate the limit
$$\lim_{n\to\infty}\sum_{k=n+1}^{2n}\frac{1}{k}$$
Can someone please explain how I can go about doing this?
- 21
3
votes
1
answer
276
views
Find limit $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^4}{k^5+n^5}$.
Find the following limit:
$$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^4}{k^5+n^5}$$
I had an idea of using upper Riemann sum for function $x^4$ on interval $[0,1]$ but I don't know how to deal with $...
- 41
0
votes
2
answers
111
views
Why isn't $\lim_{n \to \infty} \int_{1/n}^{1} \frac{1}{x}dx$ equivalent to itself rewritten as a Riemann sum?
I was comparing integrals to their equivalent riemann sums, specifically the harmonic series and I derived that:
$$\displaystyle{\lim_{n \to \infty}} \int_{1/n}^{1} \frac{1}{x}dx = \displaystyle{\lim_{...
- 19
0
votes
0
answers
102
views
Definite integrals as Riemann sums
The Riemann sum/integral, is defined to be
$$ \int_a^b f(x)dx := \lim_{n,\Delta x_i \rightarrow 0} \sum_{i=1}^n f(x_i^*)\Delta x_i $$
whenever the sum exists, where $\Delta x_i$ is the sub-interval ...
- 852
0
votes
1
answer
54
views
Determine function from Reimann sum
Determine which of the following is equal to
$$
\lim_{n\to \infty} \frac{34}{n} \sum_{i=1}^n \left(\frac{34^2i^2}{n^2} + 1 \right)
$$
(Answer: the area of the region above the $x$-axis below $f(x)=(x-...
- 270
0
votes
0
answers
82
views
How to isolate $\sum_{i=1}^{n}\frac{1}{i^{2}}$?
I have the following equation where $c$ may be some constant (derived from the Riemann sum of $\int_{1}^{x}\frac{1}{s^{2}}ds$):
$\displaystyle \sum_{i=1}^{n}\frac{1}{\left(n+i\left(x-1\right)\right)^{...
- 175
4
votes
1
answer
96
views
Justify $\int_0^\infty \frac{\sin(x)}{x}dx = \lim_{\theta\to 0}\sum_{n=0}^\infty\left( \theta \cdot \frac{\sin(n\theta)}{n\theta} \right)$
I encounter the following equation
$$\int_0^\infty \frac{\sin(x)}{x}dx = \lim_{\theta\to 0}\sum_{n=0}^\infty\left( \theta \cdot \frac{\sin(n\theta)}{n\theta} \right).$$
Intuitively, I think the limit ...
- 15.9k
2
votes
2
answers
70
views
Evaluate $\lim_{n \to \infty} \sum_{j=0}^{n} \sum_{i=0}^j \frac{i^2+j^2}{n^4+ijn^2}$
I am asked to evaluate: $$\lim_{n \to \infty} \sum_{j=0}^{n} \sum_{i=0}^j \frac{i^2+j^2}{n^4+ijn^2}$$
I am not experienced with double summations, but I tried simplifying the expression above into:
$$...
- 5,434
2
votes
1
answer
410
views
Comparing summation and integration for non monotonic function
$$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}$$
Compare the values of $P,Q,I$
I know ...
- 2,205
1
vote
0
answers
59
views
Is it true that $\lim_{n\to \infty}\sum_{i=1}^{n}\frac{1}{\sqrt{i}} - 2\sqrt{n} = 0$? [duplicate]
This problem originates from taking the limit of a Riemann Sum to evaluate an integral. In short, to help solve the limit, I used the bound
$$\sum_{i=1}^{n}\frac{1}{\sqrt{i}} \leq 2\sqrt{n}$$
Which ...
- 1,506
1
vote
2
answers
73
views
How to explain this basic summation rule: $\sum_{i = 1}^n c = n\cdot c$, where $c$ is a constant? [closed]
The parallel rule for definite integrals , namely,
$$\int_a^b c = c(b - a)$$ where $c$ s a constant, is rather intuitive, due to the fact that this number is computed in a way similar to the area of ...
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