Articles of sequences and series

Series convergence – factorial over products

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.

Using Ratio test/Comparison test

I have $\displaystyle\sum_{n=1}^{\infty}{\frac{(2n)!}{(4^n)(n!)^2(n^2)}}$ and need to show whether it diverges or converges. I attempted to use the ratio test, but derived that the limit of $\dfrac{a_{n+1}}{a_n}=1$ and hence the test is inconclusive. So I now must attempt to use the comparison test, but I am struggling to find bounds to compare it to to show […]

Integral $\int_1^2 \frac1x dx$ with a Riemann sum.

How do you find the $$ \int \dfrac{1}{x} dx$$ by using the idea of a limit of a Riemann sum on the interval [1,2]? I tried splitting the interval into a geometric progression and evaluating the Riemann sum, but i cant simplify the expression at this stage.

Calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using pen and paper

How can you calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using nothing more than a pen and pencil? Simply typing this in any symbolic calculator will give us $1/81$. I could also possibly find this formula if I was actually looking at given numbers but I have never tried working backwards. By backwards, I mean to be given the […]

General solution to Wright-Fisher model – Diploid selection

Wright-Fisher models are classical theoretical results in evolutionary biology. There are two discrete time models, one for haploid selection and one for diploid selection (the meaning of these models does not matter for the purpose of my question). My question is: What is the general solution of the below haploid selection model? diploid selection: $$p(t+1) […]

Taylor expansion of composite function

My confusion is slightly related to this question. Suppose we have two nice functions $f(x)$ and $g(x)$, how do we find Taylor series of $f(g(x))$? To be more concrete, consider $f(x^2)$. In this case, we can regard it as $f(g(x))$ where $g(x) = x^2$. One way to find the Taylor series around $1$ is just […]

Find the sum of all the integers between 1 and 1000 which are divisible by 7

How can I work this one out (with workings)? “Find the sum of all the integers between 1 and 1000 which are divisible by 7” Thanks!

prove the divergence of cauchy product of convergent series $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$

i am given these series which converge. $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$ i solved this with quotient test and came to $-1$, which is obviously wrong. because it must be $0<\theta<1$ so that the series converges. my steps: $\dfrac{(-1)^{n+1}}{\sqrt{n+2}}\cdot \dfrac{\sqrt{n+1}}{(-1)^{n}} = – \dfrac{\sqrt{n+1}}{\sqrt{n+2}} = – \dfrac{\sqrt{n+1}}{\sqrt{n+2}}\cdot \dfrac{\sqrt{n+2}}{\sqrt{n+2}} = – \dfrac{(n+1)\cdot (n+2)}{(n+2)\cdot (n+2)} = – \dfrac{n^2+3n+2}{n^2+4n+4} = -1 $ did […]

Power Series representation of $\frac{1+x}{(1-x)^2}$

Can anyone work out how to do this problem, because I’m getting an answer that close to the answer in the back of the book, but mine is off by a + 1. What I do is, I first find a representation for $1/1-x$ ( which is the integral of $1/(1-x)^2$ ) and then derive […]

How can I prove that the binomial coefficient ${n \choose k}$ is monotonically nondecreasing for $n \ge k$?

I want to prove that the binomial coefficient ${n \choose k}$ for $n \ge k$ is a monotonically nondecreasing sequence for a fixed $k$. How do I do this?