To prove: The Maclaurin series for the given function converges for all real x using remainder estimation theorem .
Explanation of Solution
Given:
Function is
Proof:
As,
So, to show that the Maclaurin series for
Now,
Using Remainder theorem, it states that
Here M and r are constants,
Here, a = 0 as we are dealing with Maclaurin series
Since
Either
Which means that
So, Set M and r equal to 1.
Then,
Since
Also
Since factorials increase faster than exponentials. This can also be done using the Ratio test.
So, conditions are satisfied
Chapter 10 Solutions
Calculus: Graphical, Numerical, Algebraic: Solutions Manual
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