Question

given a power series with divergence and convergence x values, how can you determine further convergence

Answer 1

@wmj259 i have an example to work off

Answer 2

number 38
howd it go?

Answer 3

so i know how to find an itegral of what the radius of convergence might be, but how can you tell if these values of x converge or diverge given the information

Answer 4

Approximating Functions in Polynomials?

Answer 5

9.5 of calc II

Answer 6

sequences and series

Answer 7

I have not gotten this far. I am sorry.

Answer 8

\[\lim_{n\to\infty}\left|\frac{C_{n+1}x^{n+1}}{C_nx^n}\right|=|x|\lim_{n\to\infty}\left|\frac{C_{n+1}}{C_n}\right|\]
Assuming the limit exists and is equal to \(L\), the series will converge for \(|x|<\dfrac{1}{L}\).

Given that the series is said to converge when \(x=-4\), you have \(|-4|=4<\dfrac{1}{L}\).

You're also told that the series diverges for \(x=7\), which gives you more info about \(L\). The series diverges if the limit is greater than 1, i.e. diverges when \(|7|=7>\dfrac{1}{L}\), and converges when \(\dfrac{1}{L}<7\).

So we then know that the series converges when \(4<\dfrac{1}{L}<7\).

Knowing this, we know that the series certainly converges for any values \(-4\le x\le4\), and certainly diverges for \(x\le-7\) and \(x\ge7\).

However, we don't know how long the interval is because we don't know what the value of \(L\) is, which means we can't say anything for sure about the convergence of this series for \(-7<x<-4\) or \(4<x<7\).