Articles of sequences and series

Series help, fourier series

How do I know if a given function can be represented by a fourier series, that converges to the value of that function at non discontinuities. Also where did Fourier come up with the idea of representing periodic function in the way he did?

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?