Learning Materials
Features
Discover
Chapter 3: Problem 15
Show that \(\lim \left(n^{2} / n !\right)=0\).
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
Give an example of two divergent sequences \(X\) and \(Y\) such that: (a) their sum \(X+Y\) converges, (b) their product \(X Y\) converges.
Show that if \(X\) and \(Y\) are sequences such that \(X\) and \(X+Y\) are convergent, then \(Y\) is convergent.
. Let \(\left(x_{n}\right)\) be a bounded sequence and let \(s:=\sup \left(x_{n}: n \in \mathbb{N}\right)\). Show that if \(s \notin\left\\{x_{n}: n \in \mathbb{N}\right\\}\), then there is a subsequence of \(\left(x_{n}\right)\) that converges to \(s\).
Let \(\sum_{n=1}^{\infty} a(n)\) be such that \((a(n))\) is a decreasing sequence of strictly positive numbers. If \(s(n)\) denotes the \(n\) th partial sum, show (by grouping the terms in \(s\left(2^{n}\right)\) in two different ways) that \(\frac{1}{2}\left(a(1)+2 a(2)+\cdots+2^{n} a\left(2^{n}\right)\right) \leq s\left(2^{n}\right) \leq\left(a(1)+2 a(2)+\cdots+2^{n-1} a\left(2^{n-1}\right)\right)+a\left(2^{n}\right)\) Use these inequalities to show that \(\sum_{n=1}^{\infty} a(n)\) converges if and only if \(\sum_{n=1}^{\infty} 2^{n} a\left(2^{n}\right)\) converges. This result is often called the Cauchy Condensation Test; it is very powerful.
If \(\sum a_{n}\) with \(a_{n}>0\) is convergent, then is \(\sum \sqrt{a_{n} a_{n+1}}\) always convergent? Either prove it or give a counterexample.
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine