14
Dec
2020
Question of the Day By Maths Grinds
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Will the following converge or diverge? $\sum\limits_{n=1}^\infty \frac{ 2^n x^n }{ n+1 } $
13
Dec
2020
Question of the Day By Maths Grinds
Prove that p-series test from the integral test.
12
Dec
2020
Question of the Day By Maths Grinds
The functions $h(x)$ and $g(x)$ are differentiable such that the derivative of $h$ is $h$ and the derivative of $g$ is $g$. That is $h'(x)=h(x)$ and $g'(x)=g(x)$, also, the function $h(x)$ is non-zero for all $x$. Prove that $g(x) = k h(x)$ for some constant $k$.
11
Dec
2020
Question of the Day By Maths Grinds
Where (if anywhere) is the function $f(x)=x^3-3x^2+3x-1$ decreasing? Provide a sketch.
10
Dec
2020
Question of the Day By Maths Grinds
Prove $e-1 \leq \int_{0}^{1} \left(1+x\right)^\frac{1}{2} e^x \leq \sqrt{2} \left(e-1\right) $
9
Dec
2020
Question of the Day By Maths Grinds
If the function $f$ is less than the function $g$ over some interval $a$ to $b$ and both functions are Riemann Integrable from $a$ to $b$ then prove that the integration of $f$ is also less than the integration of $g$ over the interval.
8
Dec
2020
Question of the Day By Maths Grinds
If $a_n$ is a bounded monotonically increasing sequence prove then that $b_n = \frac{a_1 + \cdots + a_n}{n}$ is also bounded and monotonically increasing.
8
Dec
2020
TIMSS Results By Maths Grinds
Two years ago this month I gave my opinion on the state of Mathematics education in Ireland. In my post I predicted that our drive towards the new Maths syllabus would cause us to increase in our PISA rankings. But this was a bit of a hack. Anyway. My opinion being that this change in... Read More
7
Dec
2020
Question of the Day By Maths Grinds
Prove that a bounded monotone increasing sequence is convergent.
6
Dec
2020
Question of the Day By Maths Grinds
Find the Taylor Series expansion of $\cosh x$ at the point $a=0$ and show that the remainder convergences to zero over $\left(-r,r\right)$ regardless the size of $r$.