# Tagged Questions

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### Using integral estimation to show that $\sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$

Show with Integral estimation that $$\sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$$ $$f(k)=\frac {\ln k}{k^2}$$ For the integral it is : 1 But the other part is the ...
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### Estimate for integral of sine to the power of $-(1+a)$ where $a>0$

I'm trying to solve or estimate this integral $$I=\int\limits_{\arcsin{k}}^{\pi/2}\dfrac{1}{(\sin{x})^{1+a}}\mathrm{d}x,$$ where $0<k<1/2$ and $a>0$. The estimate should depend on $k$. I ...
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### Performance estimation of shellSort

I'm trying to make a performance estimation for shell-sort algorithm. And I fail in it. My formula: equals to where dz is outer while-loop, dy is middle for-loop, and dx is inner for-loop ...
Let $\phi$ be continuous in a neighborhood of $0\in\mathbf{R}^3$ (you may assume it to be uniformly continuous, if you like). Do we have that $$\lim_{\epsilon\rightarrow ... 1answer 419 views ### error of the composite trapezoidal rule Let f\in C^2[a,b]. The composite trapezoidal rule is given by$$T_n[f]:=h\left(\frac{f(a)+f(b)}{2}+\sum_{k=1}^{n-1}f(x_k)\right)\;\;\;\;\;\left(h:=\frac{b-a}{n},\;x_k:=a+kh\right)$$First, I've ... 2answers 122 views ### How do we prove the error estimation of the rectangle method Let f\in C^2[a,b]. An approximation of the integral over [a,b] is given by$$I[f]:=\int_a^bf(x)\text{ dx}\approx \frac{b-a}{n}\sum_{i=1}^nf\left(a+\frac{2i-1}{n}(b-a)\right)=:M_n[f]$$I've spent ... 1answer 219 views ### Estimating the sum \sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)} By integral test, it is easy to see that$$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$converges. [Here \ln(x) denotes the natural logarithm, and \ln^2(x) stands for (\ln(x))^2] I am ... 1answer 72 views ### Some estimate concerning hyperbolic functions I want to show that |\sinh(az)|\leq|\sinh(z)| for all z\in\mathbf{C} (or at least for all z\in\mathbf{H}, the upper half plane), provided that 0<a<1 However, I am not even certain ... 1answer 112 views ### Prove  1 \lt \int_0^1 \frac{1+x^{30}}{1+x^{60}} \ \mathrm{d}x \lt 1 + \frac{1}{30} [duplicate] Possible Duplicate: How to prove \int_0^1 \frac{1+x^{30}}{1+x^{60}} dx = 1 + \frac{c}{31}, where 0 \lt c \lt 1 How can I prove the estimate$$ 1 \lt \int_0^1 \frac{1+x^{30}}{1+x^{60}} \ ...
I encountered the following in a paper and do not understand how the error term is being bounded. In what follows, $n$ and $k$ are large integer constants.  \sum_{i=0}^{n-1} \ln\left(1 - ...