Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 8: Q-37E (page 435)
Question: Let
Show that fn(0)=0for n=0,1,2....and hence that the Maclaurin series for f(x)is 0+0+0+...., which converges for all xbut is equal to f(x)only when x=0. This is an example of a function possessing derivatives of all orders (at x0=0), whose Taylor series converges, but the Taylor series (about x0 =0) does not converge to the original function! Consequently, this function is not analytic at x=0.
Short Answer
Function is not analytic at x=0 .
Step by step solution
Taylor series
For a function f(x) the Taylor series expansion about a point x0 is given by, .
The derivatives of f
Let , then f(x)=e .
For , the derivatives of f are:
f'(x)=e
f''(x)=e
f'''(x)=e
Note that f(n)(x) will be of the form ep(x), where p(x) is some polynomial.
From the definition of f it follows that, f(0)=0 .
To calculate the derivative at x=0
We need to calculate the derivative at x=0 by definition,
Over 30 million students worldwide already upgrade their learning with 91影视!
