Hello, 12MED34!
You're right . . . you need the Taylor Series.
Here's the first one . . .
If , show that: .
First, we'll crank out the Taylor Series for
Formula: .
We have: .
Then: .
We have: .
Hence: .
Therefore: . . . . . There!
I'm in an analysis class currently discussing differentiation. I'm pretty sure I need to use Taylor's Theorem on these but I'm not sure. Could someone work me through them?
First, if x is greater than zero, show that (1+x)^(1/3)-(1+(1/3)x-(1/9)x^2) is less than or equal to (5/81)x^3.
Second, if abs(x) is less than or equal to one, show that abs([sin(x)-(x-(x^3)/6+(x^5)/120]) is less than (1/5040).
Sorry. I know it's quite complicated, but I have no idea. Anyways, thank you for any help you can give.
What follows is a bit pedantic, but the OP does say that he is in an analysis
class, so I think I will assume he needs to treat the remainder in a Taylor series
explicitly.
In fact this is not a proof, you are making an assumption about the remainder
for a truncated Taylor series, expanding about :
where:
In this case , , and is a decreasing function of , which allows one to obtain a bound on the remainder:
and the conclusion follows.
RonL