You can use that $p$-series for any $p \in \mathbb{R}$ such that $$1 < p \leq 1.2$$ The resulting $p$-series will have terms $$\frac{1}{n} >a_n \geq \frac{1}{n^{1.2}}$$ And the series with ...

You know that $\cot(x) = 1 / \tan(x)$. Take random letter, let $a = \tan (x)$. Then it is $$1/a + a = 2$$ $$1 + a^2 = 2a$$ $$a^2 - 2a + 1 = 0$$ $$(a-1)^2 = 0$$ $$a - 1 = 0$$ $$a = 1$$ $$\tan(x) = 1$$ ...

If you wanted to find the probability that the first particle would appear after $4$ seconds from now, I think it would be this: $$1-\int_0^4 \lambda e^{-\lambda t} \,\mathrm{d}t$$ But you are asked ...
According to fundamental theorem of algebra, there are at least $n$ roots in complex numbers to a polynomial of degree $n$. Some of them will be real roots, some will not. At least one real root is $... View answer 0 votes Solve for$y$in terms of$x$. Then find$dy/dx$and set it equal to the slope of the tangent line (which will be -1). $$\frac{dy}{dx} = \frac{d}{dx}\left( -x + k\right) = -1$$ You should also draw a ... View answer 0 votes I found the answer from here: https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-b-chain-rule-gradient-and-directional-derivatives/session-37-... View answer 0 votes You can do this with binomial coefficients.$\binom{8}{5}$and$\binom{12}{5}$. View answer 0 votes Hint: $$(2n)(2n+1)(2n+2) = (2n+1)^3 - (2n+1)$$ View answer 0 votes You have to stand on the shoulders of giants. You cant go through life trying to discover everything yourself. Learn from what others have done in the past ... theres a reason you can learn 100 years ... View answer 0 votes Differentiability implies continuity, but the intervals$(a,b)$and$[a,b]$were not the same; the first was open second was closed. This means at the points$a$and$b$it can be not continuous and ... View answer -1 votes $$4^n - 1 = (4-1) (4^{n-1} + 4^{n-2} + \cdots + 4 + 1)$$ Each trapezoid contains$4-1$triangle View answer -3 votes Every student should know the powers of two up to$2^{20}\$ like the back of his/her hand, or in the case of a student with no hands, like some other familiar body part that would serve my rhetorical ...