Maclajrin Series

Question 1

Why cantfind maclaurin series for ln x?

Answer 1

Because look at first principles, you have to evaluate it at 0, and ln x is not possible to evaluate here so csnt

Question 2

How do we get expansion for ln (1+x)?

Answer 2

Diffenrktwte them whatever evaluate . Do the 4th INE if not obvious
What we can see is the top numbers are factorials too

Simply and cancel and you’ll see pattern

2) altenrtsign between positive snd negative, so do (-1) ^ r+1 based on first term

Question 3

Why is ln (1+x) not valid for more than 1?

Answer 3

Looking into it , this will become unbounded and so will DIVERGE away from answer !

So to it be convergent must be -1 <x<= 1

Question 4

If they give you a binomial expansionish series and they say find range of values valid?

Answer 4

Use the binomial way, and then show how mod x less than 1 etc

Question 5

How to use hack to easy expansion

Answer 5

Diffenrjtste first term if it gives you something that you can use BINOMIAL EXPANSION FOR
- expand that out
- rememebr if they say find until 5th term, only need 4th term, and integrates thst
- now remember intervstikns will give a CONSTANT . Find this CONSTSNT by subbing x = 0 in Origano expansion

Final result

Question 6

Remmeber when two expansions?

Answer 6

Find first few terms of both

Mutlilyk kut EACH TERM untuk you get your power

Keep going, don’t forget any silly term behind