maths question binomial series

hi, please could I have some help with question 5c? i'm not sure how to do it
paper: https://www.physicsandmathstutor.com/pdf-pages/?pdf=https%3A%2F%2Fpmt.physicsandmathstutor.com%2Fdownload%2FMaths%2FA-level%2FC4%2FPapers-Solomon%2Ffor-Edexcel%2FSolomon%20L%20QP%20-%20C4%20Edexcel.pdf
thanks!

Reply 1

mqb2766

Original post by anonymous56754

Youd need the values/range for x such that all the (infinte) series converge. A simple interpretation is that you cant divide by zero in the original expression so ....

Some examples at
https://www.examsolutions.net/tutorials/exam-questions-binomial-expansion/

Reply 2

anonymous56754

Original post by mqb2766

Youd need the values/range for x such that all the (infinte) series converge. A simple interpretation is that you cant divide by zero in the original expression so ....
Some examples at
https://www.examsolutions.net/tutorials/exam-questions-binomial-expansion/

Yes so can't x be any real number except -0.5 and 1

Reply 3

mqb2766

Original post by anonymous56754

Yes so can't x be any real number except -0.5 and 1

Not quite. For the simple function 1/(1+x) and its binomial series (up to x^7) are
https://www.desmos.com/calculator/jbga6wwiop
The series doesnt have a hope in hell of converging to the original fucntion for x<-1 as the series is heading off to +inf which is what the function does as it approaches x=-1 from the right (so it converges when x>-1). However the function for x<-1 is negative. A series will not approximate across a vertical asymptote.

Also, if x>1, the series will not converge as terms like +/-x^100 ... etc will head off to infinity. So it converges for |x|<1 or -1<x<1.