All Questions
14
questions
1
vote
2
answers
82
views
Upper rectangle area sum to approximate 1/x between $1\leq x\leq 3$
I am trying to figure out how to use rectangles to approximate the area under the curve $1/x$ on the interval $[1,3]$ using $n$ rectangle that covers the region under the curve as such.
Here is what I ...
- 1,071
2
votes
4
answers
273
views
How did Rudin change the order of the double sum $\sum_{n=0}^\infty c_n\sum_{m=0}^n\binom nma^{n-m}(x-a)^m$?
I see many people change the order of sum but I don't understand how they did that.
Is there is a way to change the order of the sum, $\sum\limits_{k=a}^n\sum\limits_{j=b}^m X_{j,k}$ and $\sum\...
2
votes
2
answers
295
views
$\lim_{n\to \infty}(\lim_{x\to 0}( 1+\sum_{k=1}^n\sin^2(kx))^\frac{1}{n^3x^2} )$
$$\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n\sin^2(kx)\right)^\frac{1}{n^3x^2} \right)$$
$$=\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n(k^2x^2)\frac{\sin^2(kx)}{k^2x^2}\...
0
votes
1
answer
55
views
How can $\frac1s(\sum_{k=0}^\infty \frac{s^k}{k!}-1) = \frac1s(\sum_{k=1}^\infty \frac{s^k}{k!})$?
I'm having trouble with a proof of the moment generating function via Taylor series in Introduction to probability, statistics and random processes which, in relevant part, states the following:
$$\...
- 199
1
vote
0
answers
86
views
Limit of a sum and two ways yielding two answers
$$\lim_ {n\to \infty} \sum_{r=0}^n \frac{1}{(2r+1)(2r+3)(2r+2)}$$
Now, what I did here was to first break up the general term of the sum using partial fractions, yielding
$$ \frac{1}{(2r+1)(2r+2)(2r+3)...
- 33
0
votes
2
answers
55
views
Estimating $\frac{\sum_{r = 1}^{N} r^n}{N^n}$, getting a good numerical approximaion for large $N$
I am trying to find an accurate estimate for
$$f(n,N) = \frac{\sum^{N}_{r = 1} r^n}{N^n}$$
for large values of $N$.
I know the result that as ${N\to \infty}$, the $\frac{f(n,N)}{N}$ assumes value of $\...
- 1,334
2
votes
0
answers
137
views
Can this summation be done without calculator?
Is it possible to perform the summation ,
$$\sum_{i=1}^{\infty} \frac{1}{i^i}$$
without the use of calculator?
It does converge to a finite value = 1.29129...
Wolfram Alpha link to this
Describe the ...
- 1,663
9
votes
2
answers
106
views
Finding limit for infinite quantities.
Find $$\lim_{x\rightarrow 0}{\color{red}{x}} \cdot\bigg(\dfrac{1}{1+x^4}+\dfrac{1}{1+(2x)^4}+\dfrac{1}{1+(3x)^4}\cdots\bigg)$$
As terms are written infinitely my intuitions doesn't let to give answer ...
- 1,943
2
votes
1
answer
30
views
Using a Summation or an Integral to Add Multiple Sequential Sums of an Algebraic Equation
So, as the title states, I'm trying to make a single summation or integral that will solve for the sequential sums of an already defined algebraic equation. The equation in question is this:
$$y=2x+...
3
votes
6
answers
176
views
$\lim_{x\to \infty}\left(\frac{1}{x}\sum_{i=1}^x(\frac{i}{x})^9\right)$
For precalculus I had an exam today and we had to solve the following question:
$\lim_{x\to \infty}\left(\frac{1}{x}\sum_{i=1}^x(\frac{i}{x})^9\right)$
I can't use l'hôpitals rule to solve this. ...
- 99
1
vote
0
answers
42
views
Evaluating a limit involving summation [duplicate]
Evaluate :
$$ \lim_{n\to\infty}\left(\dfrac{1}{e^{n}}\displaystyle \sum_{r=0}^{n} \dfrac{n^r}{r!}\right) $$
Numerical calculation suggests that the limit should be $\dfrac{1}{2}$. I tried using ...
- 5,719
0
votes
1
answer
213
views
Is there a summation formula for this equation (contains square roots, and functions within the square root)?
I am trying to solve a summation formula that is quite complex. However, to make the "answering" process for you guys easier I'll isolate the part I am having trouble with...
The equation is as ...
3
votes
2
answers
265
views
A limit about euler's constant
Show that :
$$\lim_{m\to \infty}\left[ -\frac{1}{2m}+\ln \left( \frac{\text{e}}{m} \right)+\sum\limits_{n=2}^m \left( \frac{1}{n}-\frac{\zeta \left( 1-n \right)}{m^n} \right) \right]=\gamma $$
How to ...
- 3,955
6
votes
7
answers
27k
views
Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$ [duplicate]
Possible Duplicate:
Value of $\sum x^n$
Proof to the formula
$$1+x+x^2+x^3+\cdots+x^n = \frac{x^{n+1}-1}{x-1}.$$
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