terrace
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True or false: There exists a series $\sum_n a_n$ of non-negative terms that is convergent, such that $\sum (a_n)^{5/6} $ diverges.
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3 votes

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 ...

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Find the general solution of equation $\cot x+\tan x=2$
Accepted answer
1 votes

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

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Finding probability a particle will appear after t seconds (exponential r.v)
1 votes

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 ...

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Existence of a nontrivial solution to a polynomial equation
1 votes

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 $...

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Determining value of k (calculus)
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 ...

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How do I find the normal vector?
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-...

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There are eight males and 12 females in a certain club. In how many ways can a committee of five be chosen if it is to consist-
0 votes

You can do this with binomial coefficients. $\binom{8}{5}$ and $\binom{12}{5}$.

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Calculate the value of the series $\,\sum_{n=1}^\infty\frac{1}{2n(2n+1)(2n+2)}$
0 votes

Hint: $$(2n)(2n+1)(2n+2) = (2n+1)^3 - (2n+1)$$

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How to get rid of "ox near amorphous mountain" feeling?
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 ...

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Function differentiable on $(a,b)$ but not continuous on $ [a,b]$
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 ...

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Induction question about triangle pattern
-1 votes

$$4^n - 1 = (4-1) (4^{n-1} + 4^{n-2} + \cdots + 4 + 1)$$ Each trapezoid contains $4-1$ triangle

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What is the sum of the prime factors of $2^{16}-1$?
-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 ...

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