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 […]

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

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

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 […]

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

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 $X_n$.

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.

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 […]

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 […]

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} […]

Intereting Posts

Unit speed reparametrization of curve
Ways to Choose Three Adjacent Elements from a Set
If $a^2$ divides $b^2$, then $a$ divides $b$
Is there any mathematical meaning in this set-theoretical joke?
Diffuse-like decomposition of the segment $$ in accordance with Lebesgue measure
How to integrate an exponential raise to the inverse sine?
What does “X% faster” mean?
Returning numbers included in a number
Mean of a Cauchy Distribution
How to prove $\prod_{i=1}^{\infty} (1-a_n) = 0$ iff $\sum_{i=1}^{\infty} a_n = \infty$?
Compute the sum $\sum_{k=1}^{\infty}k^mz^k$ where $|z|<1$
Can the the radius of convergence increase due to composition of two power series?
The Velocities of the Contact Points of Two Rolling Curves are Equal at the Instant of Contact
Simplification of expressions containing radicals
Symbolic coordinates for a hyperbolic grid?