I have to find the root of $x-e^{-x}=0$ by using fixed-point iteration. when i rewrite the equation as $x=e^{-x}$ , the iterative process converges to $0.567$ after $12$ iteration. But when i rewrite the equation as $x=-\ln(x)$ , the iterative process diverges. Why?

Does $$ \sum_{n\ =\ 1}{n!\over \left(\,\sqrt{\,2\,}\, + 1\,\right) \left(\,\sqrt{\,2\,}\, + 2\,\right)\ldots \left(\,\sqrt{\,2\,}\, + n – 1\,\right)\left(\,\sqrt{\,2\,}\, + n\,\right)}\quad $$ converge or diverge ? I used the D’Alambert criterion, but it gives $D = 1$, and I have no idea what other criterion I could use.

Conway and Guy defined $$H_k^0=\sum_{n=1}^k\dfrac1n$$ and $$H_k^r=\sum_{n=1}^kH_n^{r-1}$$ for $k,r\in\Bbb Z^+$. I would prefer a definition of an $r$-hyperharmonic number to have some chance of converging when $r>0$. Consider that the harmonic series diverges because the integers are too slow-growing, but the harmonic numbers also begin at $1$ and grow much more slowly, so $$\sum_{n=1}^kH_n^0$$ doesn’t […]

I’m trying to decide if these series converge or diverge: $$\sum_{n=1}^{\infty} (-1)^n \left(\frac{2n + 100 }{3n + 1 }\right)^n $$ Here $\lim_{n\to\infty} \left(\frac{2n + 100 }{3n + 1 }\right)^n \ne 0$, so can we conclude that the series diverges? $$ \sum_{n=1}^{\infty} \log \left( n \sin \frac{1}{n} \right) $$ Can we compare this series with the […]

So, in the classic arcade game Asteroids, you move in a game field where the top and bottom edges are identified and the left and right edges likewise, topologically a torus. I’m interested in how such a game world might accommodate gravity. One model I might use is to tile the plane with copies of […]

This question came to me from one of my calculus students today: Other than using the integral test $$\int_1^\infty \frac{dx}{x} \to \infty,$$ what are some other ways that we can prove the Harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ diverges? I’m sure there are plenty of methods out there; any method where a typical student in Calculus could […]

I understand that the Cesaro summation is a way of assigning values to divergent sums based on the limits of averages. Are there other approaches to assigning values? In the case of $1+2+3+4\dots$, there is no limit to the average. So, if I am understanding right, Cesaro Summation does not assign a value for this […]

Does $ 1 + 1/2 – 1/3 + 1/4 +1/5 – 1/6 + 1/7 + 1/8 – 1/9 + …$ converge? I know that $(a_n)= 1/n$ diverges, and $(a_n)= (-1)^n (1/n)$, converges, but given this pattern of a negative number every third element, I am unsure how to determine if this converges. I tried to […]

Need help finding a minorant to $(\sqrt{k+1} – \sqrt{k})$ which allows me to show that the series $\sum_{k=1}^\infty (\sqrt{k+1} – \sqrt{k})$ is divergent.

I saw $\zeta (1/2)=-1.4603545088…$ in this link. But how can that be? Isn’t $\zeta (1/2)$ divergent since $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+..>\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+..$ ?

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