Suppose $\displaystyle f(x,\alpha )=\begin{cases}
\dfrac{e^{-\alpha x}-e^{-2\alpha x}}{x} & \text{ if } x\neq 0 \\\\
\alpha & \text{ if } x=0
\end{cases}$.
Is $\displaystyle \int_{0}^{\infty }f(x,\alpha )dx$ uniformly convergent for all $\alpha \geq 0$ ? If yes, then how? Please help.
Search found 14 matches
- Sat Aug 13, 2016 12:38 pm
- Forum: Real Analysis
- Topic: Uniform Convergence of an Improper Integral
- Replies: 1
- Views: 2394
- Wed Jul 06, 2016 6:37 pm
- Forum: Calculus
- Topic: An interesting integral!
- Replies: 2
- Views: 2305
Re: An interesting integral!
Thank you!!
- Sat Jun 25, 2016 5:17 pm
- Forum: Calculus
- Topic: An interesting integral!
- Replies: 2
- Views: 2305
An interesting integral!
\[ \int_{0}^{\infty }\frac{\ln^{2}x}{x^2+2x\cos\theta+1 }dx=?\ where\ \theta\in [0,\pi ] \]
- Sat Jun 25, 2016 3:12 pm
- Forum: Calculus
- Topic: \( \int_{0}^{\infty }\left \lfloor{2016\,e^{-x}}\right \rfloor dx=? \)
- Replies: 0
- Views: 1398
\( \int_{0}^{\infty }\left \lfloor{2016\,e^{-x}}\right \rfloor dx=? \)
\[ \int_{0}^{\infty }\left \lfloor{2016\,e^{-x}}\right \rfloor dx=?
\]
\]
- Mon Jun 20, 2016 2:59 pm
- Forum: Calculus
- Topic: A series with factorials in the denominator
- Replies: 3
- Views: 2705
Re: A series with factorials in the denominator
By using Discrete Fourier Transform (used in the http://www.mathimatikoi.org/forum/viewt ... f=27&t=686 )
\(\mathcal{S}(x)=\sum_{n=0}^{\infty} \frac{x^{4n}}{(4n)!}\)
\(= \frac{1}{2}\left ( \cosh x +\cos x \right )\)
\(\mathcal{S}(x)=\sum_{n=0}^{\infty} \frac{x^{4n}}{(4n)!}\)
\(= \frac{1}{2}\left ( \cosh x +\cos x \right )\)
- Wed Jun 15, 2016 6:28 pm
- Forum: Calculus
- Topic: Interesting Series
- Replies: 5
- Views: 3881
Re: Interesting Series
Simply Beautiful Solution!!!
Thank you soooo much!
Thank you soooo much!
- Wed Jun 15, 2016 2:51 pm
- Forum: Calculus
- Topic: Interesting Series
- Replies: 5
- Views: 3881
Interesting Series
\[\sum_{{n=0}}^{\infty} \frac{1}{\left ( 3n \right )!}= ?\]
- Thu Mar 24, 2016 3:27 pm
- Forum: Calculus
- Topic: $\sum_{n=2}^{\infty}(-1)^n \frac{\ln n}{n} =?$
- Replies: 2
- Views: 3620
$\sum_{n=2}^{\infty}(-1)^n \frac{\ln n}{n} =?$
$$\sum_{n=2}^{\infty}(-1)^n \frac{\ln n}{n} =?$$
- Sat Feb 13, 2016 9:45 pm
- Forum: Calculus
- Topic: Challenging Elliptic Integral
- Replies: 0
- Views: 1405
Challenging Elliptic Integral
A Problem by a Great Teacher and Mathematician!
Solve using Elliptic Integrals:
$$ \int_{0}^{a}\frac{1}{\sqrt{(\sin x+\cos x)(1+2\sin 2x-4\sin^22x)}} dx=? $$
Solve using Elliptic Integrals:
$$ \int_{0}^{a}\frac{1}{\sqrt{(\sin x+\cos x)(1+2\sin 2x-4\sin^22x)}} dx=? $$
- Sun Jan 17, 2016 11:38 pm
- Forum: Calculus
- Topic: trig integral with divergent pieces, but converges together
- Replies: 2
- Views: 2623
Re: trig integral with divergent pieces, but converges toget
We have that: $$\int \left [ \left ( \frac{3}{x^2}-1 \right )\sin x - \frac{3}{x}\cos x \right ]^2 \, dx = \int \left ( \frac{9}{x^4}\sin^2 x +\sin^2 x - \frac{6}{x^2}\sin^2 x + \frac{9}{x^2}\cos^2 x -\frac{18}{x^3}\sin x \cos x + \frac{3}{x}\sin 2x \right )\, dx$$ Using Tabular Integration, we have...