Articles of sequences and series

Divergence of the serie $\sum \frac{n^n}{n!}(\frac{1}{e})^n$

Show that the serie $$\sum \frac{n^n}{n!} \big(\frac{1}{e}\big)^n$$ Diverges. The ratio test is inconclusive and the limit of the term is zero. So I think we should use the comparasion test. But I couldnt find any function to use, I’ve tried the harmonic ones, but doesnt work, since I cant calculate the limits. My guess is […]

Prove that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +…+ \frac{n}{2^n} = 2 – \frac{n+2}{2^n} $

I need help with this exercise from the book What is mathematics? An Elementary Approach to Ideas and Methods. Basically I need to proove: $$\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +…+ \frac{n}{2^n} = 2 – \frac{n+2}{2^n} $$ $i)$ Particular cases $ Q(1) = \frac{1}{2} ✓ $ ———- $ P(1) = 2 – \frac{1+2}{2^1} = \frac{1}{2} ✓ $ $+ = 1 […]

Show root test is stronger than ratio test

This question already has an answer here: Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$ 1 answer

All $11$ Other Forms for the Chudnovsky Algorithm

Continued from this post Ramanujan found this handy formula for $\pi$$$\frac 1\pi=\frac {\sqrt8}{99^2}\sum_{k=0}^{\infty}\binom{2k}k\binom{2k}k\binom{4k}{2k}\frac {26390k+1103}{396^{4k}}\tag1$$ Which is related to Heegner numbers. Sometime after, the Chudnovsky brothers came up with another $\pi$ formula$$\frac 1\pi=\frac{12}{(640320)^{3/2}}\sum_{k=0}^\infty (-1)^k\frac {(6k)!}{(k!)^3(3k)!}\frac {545140134k+13591409}{640320^{3k}}\tag2$$ And according to Tito, $(2)$ has a total of $11$ other forms with integer denominators. Question: What are all $11$ […]

Lagrange inversion theorem application

Can someone give me an example of where the Lagrange inversion theorem is applied in such a way it inverts a formal series? For example, say I have $$\sum_{i>-1} a_it^i = u.$$ Can someone show me the step by step process by which $$\sum_{i>-1}b_iu^i = t$$ is obtained. I can seem to find any links […]

Estimating the series: $\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$

Any idea on how to estimate the following series: $$\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$$ where $a$ and $b$ are constant values. Greatly appreciate any respond.

Let $a>0$ and $x_1 > 0$ and $x_{n+1} = \sqrt{a + x_n}$ for $n \in \mathbb{N}$. Show that $\{x_n\}_{n\ge 1}$ converges

This question already has an answer here: Show that the following inductively defined sequence converges and find its limit. [closed] 3 answers

What is the reasoning behind why the ratio test works?

The ratio test says that if we have $$\sum_{n=1}^{\infty}a_n$$ such that $\lim_{n \to \infty} \dfrac{a_{n+1}}{a_n} = L$, then if: 1) $L < 1$, then $\sum_{n=1}^{\infty}a_n$ is absolutely convergent, 2) $L > 1$, then $\sum_{n=1}^{\infty}a_n$ is divergent, and 3) $L = 1$, then the ratio test gives no information. I want to understand the mathematics behind […]

How to prove that the numeric series $S := \sum_{n=0}^{\infty} x^n=\frac{1}{1-x}\text{ for any } x<1$

This question already has an answer here: How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6\ldots$” to infinity = “$-1/12$”? [duplicate] 4 answers

On a “coincidence” of two sequences involving $a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$

This was inspired by this post. Define, $$a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$$ $$b_n = \sum_{k=0}^n \binom{-\tfrac{1}{2}}{k}\big(-\tfrac{1}{2}\big)^k$$ where $_2F_1$ is the hypergeometric function and binomial $\binom n k$. The first few numerators $N$ are, $$N_1(n) = \color{brown}{1, \,5, \,43, \,177, \,2867, \,11531, 92479}, \,74069, 2371495,\dots$$ $$N_2(n) = \color{brown}{1, \,5, \,43, \,177, \,2867, \,11531, 92479}, \,370345, 11857475,\dots$$ respectively, and […]