Articles of sequences and series

Showing summation is bounded

I’m currently taking a Comp Sci class that is reviewing Calculus 2. I have a question: Show that the summation $\sum_{i=1}^{n}\frac{1}{i^2}$ is bounded above by a constant I realize that this question is already answered here Showing that the sum $\sum_{k=1}^n \frac1{k^2}$ is bounded by a constant Could anyone explain it to me further? I […]

Prove that if $\sum a_n$ converges, then $na_n \to 0$.

This question already has an answer here: If $(a_n)$ is a decreasing sequence of strictly positive numbers and if $\sum{a_n}$ is convergent, show that $\lim{na_n}=0$ [duplicate] 3 answers

Find the sum of the series $\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\cdots$

This question already has an answer here: Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} – \frac{1}{2} + \cdots $ 7 answers

Theorem 3.37 in Baby Rudin: $\lim\inf\frac{c_{n+1}}{c_n}\leq\lim\inf\sqrt{c_n}\leq\lim\sup\sqrt{c_n}\leq \lim\sup\frac{c_{n+1}}{c_n}$

Here’s Theorem 3.37 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: For any sequence $\{c_n\}$ of positive numbers, $$\lim_{n\to\infty} \inf \frac{c_{n+1}}{c_n} \leq \lim_{n\to\infty} \inf \sqrt[n]{c_n},$$ $$ \lim_{n\to\infty} \sup \sqrt[n]{c_n} \leq \lim_{n\to\infty} \sup \frac{c_{n+1}}{c_n}.$$ Now Rudin has given a proof of the second inequality. Here’s my proof of the first. Let $$\alpha […]

Prove $\sum_{k=1}^{\infty} \frac{\sin(kx)}{k} $ converges

How to prove $$\sum_{k=1}^{\infty} \frac{\sin(kx)}{k}$$ converges without using integral test?

Rewriting sequence from $X_{n+1}$ to $X_n$

I have the sequence: $$ \begin{align} X_{n+1} &= \frac{X_n^2 + 5}{2X_n} \\ X_1 &= 1 \end{align} $$ I have to prove that it converges and find its limit after I write it in terms of $X_n$, which I can do, but I can’t seem to convert $X_{n+1}$ in terms of $X_n$.

Changing limits in absolutely convergent series

Let $\sum_{n=0}^\infty f(n,m)$ be a real series. Suppose the series converges absolutely. Can we do the following? $$ \lim_{m\to\infty}\sum_{n=0}^\infty f(n,m)=\sum_{n=0}^\infty \lim_{m\to\infty}f(n,m)\quad $$ I thought about some series, but all of them fit. Is there any counterexample? We suppose that all limits exist.

How to prove that $\sum_{n=1}^\infty{n^2a^{n-1}}=\frac{1+a}{(1-a)^3}$

I want to prove that $$\sum_{n=1}^\infty{n^2a^{n-1}}=\frac{1+a}{(1-a)^3}$$ I start off at the sum and try to work my way into the equation. I know that the sum is: $$σ_{n}=1+2^2+a+3^2a^2+4^2a^3+\dots+n^2a^{n-1} (1)\Leftrightarrow$$ $$aσ_{n}=a+2^2a^2+3^2a^3+\dots+n^2a^n (2)$$ If I subtract (2) from (1), I get: $$(1-a)σ_{n}=1+2^2a+3^2a^2+\dots+n^2a^{n-1}-a-2^2a^2-\dots-n^2a^n \Leftrightarrow$$ $$(1-a)σ_{n}=(n^2+(n-1)^2)a^{n-1}-n^2a^n \Leftrightarrow$$ $$σ_{n}=\frac{n^2-(n-1)^2-n^2a^n}{1-a} \Leftrightarrow$$ $$σ_{n}=-\frac{1+n^2a^n}{1-a}$$ and that is what I’ve got so far. How […]

Summations involving $\sum_k{x^{e^k}}$

I’m interested in the series $$\sum_{k=0}^\infty{x^{e^k}}$$ I started “decomposing” the function as so: $$x^{e^k}=e^{(e^k \log{x})}$$ So I believe that as long as $|(e^k \log{x})|<\infty$, we can compose a power series for the exponential. For example, $$e^{(e^k \log{x})}=\frac{(e^k \log{x})^0}{0!}+\frac{(e^k \log{x})^1}{1!}+\frac{(e^k \log{x})^2}{2!}+\dots$$ Then I got a series for $$\frac{(e^k \log{x})^m}{m!}=\sum_{j=0}^\infty{\frac{m^j \log{x}^m}{m!j!}k^j}$$ THE QUESTION I believe that we […]

Closed-form formula to evaluate $\sum_{k = 0}^{m} \binom{2m-k}{m}\cdot 2^k$

Inspired by this question I’m trying to prove that $$\lim_{m \to \infty} \sum_{k = 0}^{m} \frac{m! (2m-k)!}{(m-k)!(2m)!}\frac{x^k}{k!} \approx e^{\frac{x}{2}}$$ So I needed to find the value of $$\frac{\lim_{m \to \infty} \sum_{k = 0}^{m} \frac{m! (2m-k)!}{(m-k)!(2m)!}\frac{x^k}{k!}}{e^{\frac{x}{2}}} = \frac{\lim_{m \to \infty} \sum_{k = 0}^{m} \frac{m! (2m-k)!}{(m-k)!(2m)!}\frac{x^k}{k!}}{\lim_{m \to \infty} \sum_{k = 0}^{m} \frac{\frac{x}{2}^k}{k!}} \\ = \lim_{m \to \infty} […]