Articles of sequences and series

Mutually Exclusive Events (or not)

Suppose that P(A) = 0.42, P(B) = 0.38 and P(A U B) = 0.70. Are A and B mutually exclusive? Explain your answer. Now from what I gather, mutually exclusive events are those that are not dependent upon one another, correct? If that’s the case then they are not mutually exclusive since P(A) + P(B) […]

Finding the Sum of a series $\frac{1}{1!} + \frac{1+2}{2!} +\frac{1+2+3}{3!}+…$

I need to find the sum of this series $$\dfrac{1}{1!} + \dfrac{1+2}{2!} + \dfrac{1+2+3}{3!}+\ldots$$ But somehow I am not even convinced this converges. I tried writing it as $\sum \dfrac{n(n+1)}{2(n!)}$. I also tried to rewrite it as a telescoping series. But I seem to be stuck. Some hints on where I am going wrong would […]

Measuring the degree of convergence of a stochastic process

Consider a set of random variables $(X_1,X_2,X_3,…X_k)$ that are i.i.d. $Bernoulli(p)$ While I do not know $p$, I can estimate it using $$ Y(k)=\frac{1}{k}\sum_{i=1}^k X_i $$ Notice that $Y(k)$ is a random variable, and its distribution has mean $p$ and variance $\frac{p(1-p)}{k}$. So $Y(k)$ is a consistent estimator of $p$. Question: How can I determine […]

Convergence and uniform convergence of a sequence of functions

Suppose that ($f_n$;$n \geq 1$) is a sequence of functions defined on an interval [a,b].We will say $f_n$ tends to $f$ uniformly on $[a,b]$ as n tends to infinity if for every $\epsilon>0$ there exists a $N$ so that; $n \geq N$ implies $|f_n(x)-f(x)|<\epsilon$ for all $x \in [a,b]$. $(i)$ Consider the functions $f_n:[0,2] \rightarrow […]

Using little-o notation to prove series convergence

I think I have the correct answer to the problem specifically, but I’m a little unsure of my usage of the little-o notation, or if this is even what Apostol was suggesting to do. (Apostol does cover little-o with regard to Taylor series, he hasn’t mentioned big-O yet). The question in the book (Apostol Calculus […]

Evaluate series of $\sum_{n=1}^\infty \frac{1}{n^2+n}$

I am trying to evaluate this sum, I know that $\sum\limits_{n=1}^\infty \dfrac{1}{n^2+n}$ is called telescopic series: $$\sum_{n=1}^\infty \frac{1}{n(n+1)}$$ and I can show that as: $$\frac{1}{k}-\frac{1}{k+1}$$ I would like to get some hint how I can evaluate it. Thanks!

Leibniz's alternating series test

the definition of this test is: if $a_n$ decreases monotonically and goes to 0 in the limit then the alternating series $\sum_{n=1}^{\infty}(-1)^na_n$ converges my question is: why does the series $a_n$ has to be monotonically descending, isn’t it enough for $\lim_{n\to\infty}a_n = 0$ ? can someone give me an example for that?

Problems with a proof that every real sequence has a monotonic subsequence

I have some problems with the visualization of this proof. (I will present the problems I have with it in the end, with some intuitive thoughts related to them in italics.) Theorem: Every real sequence has a monotonic subsequence. Proof (Thurston): Take any $(x_m) \in R^\infty$ and define $S_m := \{x_m, x_{m+1},\dots \}$ for each […]

Sufficient conditions to guarantee Cesaro summarble

Is is true that every nonnegative, bounded series in $R$ is Cesaro summarble? Is there a list of sufficient conditions on series to guarantee Cesaro summarble?

What is the expansion in power series of ${x \over \sin x}$

How can I expand in power series the following function: $$ {x \over \sin x} $$ ? I know that: $$ \sin x = x – {x^3 \over 3!} + {x^5 \over 5!} – \ldots, $$ but a direct substitution does not give me a hint about how to continue.