Question

Which of the following are true statements?

All Taylor series are power series.
Taylor polynomials and Taylor series are the same thing.
No Taylor function exists that exactly equals f(x) = sin(x).

Answer 1

I only
II only
I and III only
I and II only

Answer 2

Taylor series is a special case of power series with
\[a_n=\dfrac{f^n(c)}{n!}\]

Answer 3

1 is right
2 is i guess|
3 debatable

Answer 4

I'm inclining toward
I and III only

Answer 5

like if u need an infinite number of terms to get sinx.. then is that really saying it exists

Answer 6

For any finite \(n\), the remainder is never \(0\), so \(T_n(x)\) never identically equals \(\sin(x)\)

Answer 7

ya id go with i and iii only too

Answer 8

\[T_n(x) = \sum_{i=0}^{n} \frac{ f^{(i)}(a) }{ i! }(x-a)^i\] Tn is a polynomial of degree n, so I don't think II is right

Answer 9

We can also use f(x) = e^x and check the result

Answer 10

That's a bit insulting ganeshie! Haha, but it's funny! XD

Answer 11

i better delete it before kids come to this thread

Answer 12

Haha good idea

Answer 13

For anyone that was wondering, I picked I and III and it said it is wrong. Now 3 of these answers have I in it, so either there are some Taylor series that aren;t power series (but that is not true, Taylor is just a subset) or there is a difference between the taylor series and the taylor polynomials

Answer 14

whats the definition of "taylor function" that your professor is using ?

Answer 15

Idk. The people that make the tests are a different company than the lessons (this is online)

Answer 16

Okay the question depends on the definitions we use for :
1) taylor series (clear)
2) taylor polynomial (almost clear)
3) taylor function (a bit ambiguous)