Articles of sequences and series

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.

Why Does The Taylor Remainder Formula Work?

I’ve been studying calculus on my own and have come across Taylor series. It is very intuitive until I came across the remainder part of the formula where things got fuzzy. I understand why the remainder exists but not the mathematical description. Why is the value being plugged into the derivative of the remainder some […]

zeros of linear recurence sequences

Given a linear recurrence sequence $\{a_n\}_{n\geq 0}$, how to decide whethere there are infinitely many zeros, or there are only finitely many ones?

Prove that $|a_{n+1}-a_{n}| \leq \frac{1}{2}|a_{n}-a_{n-1}|$ implies $\{a_n\}$ bounded and convergent.

I am trying to prove that every sequence for which $|a_{n+1}-a_{n}| \leq \frac{1}{2}|a_{n}-a_{n-1}|$ is true, is bounded and convergent. I tried defining $b_n = a_n$ and $c_n = a_{n-1}$ and then use Stolz, but it didn’t work. How could I prove this?

Bound of sequence

$$a_n=\frac{n\cos(n)+\sin(n^2+2n-5)}{n^2}$$ I am trying to find the bound of this sequence.But I get stuck after some computations. I take $\sin f(x)\leq|f(x)| $ and $|\cos(x)|\leq1$ but I am not sure what to do at the end.Wasted 10 papers.

The sum $\sum_{k=0}^\infty (k-1)/2^k$ is zero

I am trying to prove an equation given in the CLRS exercise book (AP GP clrs appendix A.1-4). The equation is: $$\sum_{k=0}^\infty (k-1)/2^k=0$$ I solved the LHS but my answer is 1 whereas the RHS should be 0 Following is my solution: Let’s say S = k/2^k = 1/2 + 2/2^2 + 3/2^3 + 4/2^4 […]