Using Taylor's Theorem and big O

SMA_01

1. The problem statement, all variables and given/known data

Use Taylor's Theorem to show that

√(x+1)=1+(1/2)x+O(x²)

for x sufficiently small.


Here's what I did:

f(x)= √x+1

f'(x)= (1/2)(x+1)^(-1/2)

Then using x₀=0,

f(0)= 1, f'(0)=1/2.

√x+1=1+(1/2)x-(1/8)x²(cx+1)^(-3/2)

So, then using h as a parameter:

l√(h+1) -1-(1/2)h l ≤ 1/8(h²)

Finally,

√(h+1) = 1+(1/2)h+O(h²)

Is this correct?

I' m having difficulty understanding the meaning of O, can someone please explain in simple terms?
Thank you.

LCKurtz

If you expand ##f(x)## in a Taylor series about ##x=0## you get$$
\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$$You need to write out the first few terms of that for your function. The notation ##O(x^2)## means the other terms are bounded by a constant times ##x^2##. So you group all the higher order terms together and call them ##O(x^2)##.

SMA_01

LCKurtz- Thank you. I just edited my original question, I think I solved it right, but am not completely sure. How do I know when to stop expanding the series? Is it once I get x^2?

LCKurtz

Yes, the terms from ##x^2## and higher are all ##O(x^2)##. I like to think of ##O(x^2)##, very informally mind you, as "Other terms containing ##x## to at least second power". Not very precise but easy to remember.

SMA_01

Much easier to understand, thank you!